second time puzzled.

on nerdiness

2010-05-02T03:25:00.000-07:00

This generation has grown to celebrate nerdiness. Popular media has glorified nerdiness no end. In the nineties, fed on cable tv, my idea of a nerd used to be Boris Grishenko, the evil nerd in the movie ‘golden eye’ thumbing and spinning his pen nervously while using left hand to find access codes. He has been my primary reference ever since. Many aspiring nerds, including me, used to spend hours trying to solve futile needle-in-a-haystack problems like poring over assembly code to find the instructions where zero flag should be flipped to get that app to work inside a debugger. I also learnt to play with bash shell, emacs and even some 'ed' and loved to hate user-friendly guis (but still used them when nobody was around), while going easy on the evil part. Yes there was also that character in the movie 'swordfish' but the movie or the characters sonamehow didn’t catch my imagination.

Things have changed since then. Nerds are not necessarily evil. There are benign self-loving nerds like Sheldon in bbt. There are much more benign nerds like lennard and his friends. Nerds pastimes are no longer relegated to the computer. This brings us to some questions we may ask. Can normal people be nerds? Should one aspire to be? Is it easier now to enter the ranks?

As for the last question, I think that it easier to be nerd than it is to be rocket scientist. But it is complicated. I dont think there is a set formula. Some personality types such as those with mild aspergers or a minor obsessive more easily fit the bill than others. If you are thin, bespectacled and growing a minor beard, and have shabby hair, you have made some progress but hardly enough. There are scattered data about what makes a uber nerd. But most people know a nerd when they meet one: Adam Savage for me is way up there among benign nerds. His nerdiness is infectious and we like him for that. He also shows what the very essence of being a nerd comes from his being obsessive, meticulous about his pastimes and a willingness to concentrate on some gruesome nerdy work for hours together. Biologically perhaps this has to do with a dominant right brain. But certainly, one does not need to be evil or cruel.

Some people take to superficial things such as loving themselves and being dismissive of other people, but won’t do the hard part of getting to work on their nerdy pastimes. Some spend time sharpening the pencils as opposed to doing research, upgrading the computer with latest app, learning the new trick to switch buffers in emacs, trivializing others work or being a general bully. A nerd can be excused for being him/herself only if he/she keeps the standards and quality of work high enough. Otherwise, he is just a bully who is to be ignored. How do you stand a Sheldon like character otherwise? Nerds thrive on praise by others( remember how boris flushes when natalya says, you are such a geek!) and that is earned at a price. .

As for the second question, there are social contexts where I think it is dangerous to be a nerd (even benign). Imagine being a nerd like Sheldon with queer pastimes in Afghanistan!. You would be shot at sooner or later for your remarks. That said, I guess, ones social background, position and the company one keeps should all matter in making the choice.

Nerds are not necessarily intellectuals or even wise. In fact in most cases they are plain clever with a command in an arcane subject matter which usually goes with mild to acute social maladroitness. This gets them into trouble and they often make poor choices in choosing their friends and partners and also often make poor career decisions (includes working for villains). Nerds need not be cruel but many a times we see them go to great lengths in asserting the superiority of their work while dismissing work by others as trivial ( Boris' dismissal of Natalya as second level programmer ) but one thing nerds don’t do is call themselves nerds. This is about as close to being self-effacing as they can get. They don’t want to be bracketed ever, lest they lose their uniqueness. So people who have answered a questionnaire in a website and announce to the world they are nerds are never that.

An apprentice nerd usually mimicking his idol motivates himself thinking how cool his/her work might look and nothing else but over the years the nerds learn that they cannot sustain their nerdiness unless they find an inner reservoir of strength and sustenance. They realize that that the sooner one starts relishing and performing well in ones nerdy pastime the better. The nerdiness must come from within, not by the thought of how others might see your work.

To conclude I say this: We constantly look to find things about us which are unique, and want to be believe that we are in some ways chosen ones. This often inspires us to do things out of the ordinary. Nerds entertain that delusion for long and somewhat magically keep getting better at their work and do some spectacular work. Others dismiss this as a delusion and submit to being normal human beings and are happier for that. So who do you want to be?

Git

2009-12-14T07:14:00.000-08:00

Git is fast becoming a scm tool of choice for most online development projects( Linux kernel uses it!). There are some excellent screencasts (pity, some require you to pay) that teach you how to work with it. I recently got hooked to git myself and was charmed immensely with the functionality it provided, esp the ability to have your github on web. But getting git (msysgit in my case ) to work behind proxy proved somewhat nightmarish in my home computer running windows. There are some links scattered to help you with it. After struggling for about 4 hours, I got it working using the steps outlined in link .

start msys_git.sh


$ mkdir ~/.ssh

$ chmod 700 ~/.ssh

Create public key/private key pairs using ssh-keygen

$ ssh-keygen -q -f ~/.ssh/id_rsa -t rsa

id_rsa.pub contains the public key. Copy this to git hub creation page.

Steps below are from here

start puttygen ( which should be in same folder as putty ): import the file id_rsa and convert it to create a .ppk file in the same directory.

Start putty.exe
Set the host to ssh.github.com, port 443.

Under Connection / SSH / Auth, select your private key file (.ppk file) for SSH authentication.

Under Connection / Proxy, select HTTP and enter appropriate information for the host and port.

Save your session with a name such as “gitproxy”.

Click “Open” - you should receive a login prompt. Close this.

Start pageant, the putty authentication agent. It appears on sys-tray. Rightclick and load the private key used above (.ppk file).

Back on msys_git command prompt run:


$ git remote add github git@gitproxy:svark/notes.git

$ git config --global user.name "Your Name"

$ git config --global user.email "youremail@gmail.com"

$ mkdir notes

$ cd notes

$ git init

$ touch README

$ git add README

$ git clone http://github.com/svark/notes.git

put plink in the path and make sure we have pagent still running.

$ declare -x GET_SSH="plink".

$ git commit -m 'first commit'

$ git remote add github gitproxy:svark/notes.git

$ git push github master

lonely planet.

2009-01-25T20:26:00.000-08:00

This pic of an eclipse taken in 1992 and available at nasa's site, I recently saw, struck me with some awe. It is one of those moments when Nature beckons to us, almost shyly, to leave our quibbles aside for a moment and appreciate its majestic presence. The Sun in horizon is on this occasion shrouded by a disc. Yes, it is the moon playing a little trick on us, but we can not help but feel a sense of disbelief as we behold this spectacle. Certainty of events like the sun's appearance on horizon which we take so much for granted is called into question, if only temporarily, reminding us of our own impermanence, of our vulnerability among other things.

But that cluster of palm trees also standing against Suns path seems to have a different story to tell. Life in this painfully lonely planet in an indifferent universe, is fragile and prone to destruction by cosmic elements and yet, it has kept its flag flying, defying nature and has stood tall like the cluster of palm trees in picture. The pic seemed to me a perfect ode to life on our planet, though a mute one. It mocks us at our shattered lives with its unreasonable grit to survive, taking on natures apathy in its own way. Sentimental wish-wash aside, it definitely feels good to forget all for a brief while and gaze at this celestial wonder.

katti meethi aankon ki boliyaan

2008-07-13T00:09:00.000-07:00

Bhupinders long sonorous droning. Talks and laughter in between to relax the notes, flexing rules of rendition. A deft handling of a harkat, leaving a trailing drone. I almost succumb to the drone were it not for the sharp upping of the tempo and pitch, as the sweet, almost cloying, voice of sulakshana pandit asserts itself. Notes gather momentum, but his resonant voice intervenes to pull it back:
गूँज रहे हैं दुब्द्ते सायें, शाम की खुशन्न्न्बू हात न आये..
Kushboo becomes Kkusshhhnnnbooo as if rendering with a nasal tinge this way would make it smell better. Then continues

गूंज्थी आन्खों की नशीली बोलियाँ. कट्टी मीठी आन्खोँ की रसीली बोलियान्न्न।

I close my eyes and imagine what the synesthetic experience of tasting intoxicating words from salty eyes would sound like.

That √2 is irrational

2008-02-17T05:04:00.001-08:00

Rudin's text on mathematical analysis provides a proof that √2 is irrational by showing that the set of rational numbers A consisting of all rational numbers whose square is greater than 2 has no least element and the set B of all rational numbers whose square is lesser than 2 has no greatest element. He considers A first and shows that for any p in A, we can find
a number q that is lesser than p and yet in A. He conjures up from apparently nowhere the rational number

q = p – (p^2 - 2)/(p + 2) ……….(1)

The number q is immediately seen to satisfy the needs of the proof for the set A.

How did he arrive at (1)? Some algebra shows that it does not matter what denominator is chosen in (1) as long as it is a rational greater than p + √2. So this could have been 2p for example. This it turns out is sufficient for us to work up a number such as q in (1) ourselves.

Since p > √2, we expect 2/p < √2. So √2 and hence the rational q we are searching for lies between p and 2/p, and of course q should be greater than √2. But first let us find out by what ratio the number √2 divides the segment (2/p,p). Let us say the ratio is µ. Then

√2 = μ 2/p + (1 - μ) p = μ (2/p-p) + p……….(2)

From which, we get = p / (√2 + p). The above equation suggests that as μ varies between 0 and p / (√2 + p), the number μ 2/p + (1 - μ) p spans the segment (√2,p), which is exactly the interval in which we are looking for rational numbers q. We now can find q, by choosing a appropriate value for μ. Now since p >
√n, we can choose μ = p / (p + p) < p/(√2 + p), so we have q = p/2 + 2/2p = p/2 + 1/p.

How close is this number to the in (1)?. As I said earlier in the post this would be the same number if we replaced the denominator in (1) by (p+p) instead of p + 2.

mini-supernova moment

2007-09-02T05:41:00.001-07:00

Daybreak was not a regular event in planet mersus. As it charts along its orbit through heavenly space-time, the event marks a departure in the evolutionary progress of many life-forms and for a good many others the event brings sudden extinction. As the star rays shine across dark side side of the planet, the light of a thousand suns brings a protozimmer back to its animated self; the glow all but managing to pierce its vestigial eyes long blinded by the dark ages. Yet, those eyes could feel the glow that turned the tangible to the visible. It fritters about sensing danger anticipating the unthinkable, seeking the safety of darkness, again. In an instant all hell breaks lose. Breathing in all the atmospheric helium, it emits a shriek that splits helium to puffs of hydrogen and I awake, sneezing, to find myself on a monday morning again- my mother dusting the bed.

flight to heathrow

2006-09-01T03:09:00.000-07:00

The recent Heathrow bomb scare reminded me of my flight to the U.K in march this year and a minor scare we had then. In Dubai, as I waited for the plane to take off for Heathrow, I tried to occupy myself by reading a book. Minutes before the flight took off, a 30-ish South-Asian looking guy got on board and walked to my seat. I couldnt help notice that his newish jacket was too big for his arms and his pants were short enough to make his shins visible above the socks. He had neither removed his sunglasses nor the "jim courier" cap, whilst looking around intently around my seat comparing the seat number on his boarding pass with the confusing array of A B C's and digits printed on the baggage shelf. A while later he mustered courage to ask for help in solving this number puzzle and it didnt take long before we managed to get him seated. He thanked me profusely in an accent that was distinctly British and introduced himself as Mr Latif. ( that explains why arrived so late) and said he was going back to London after his long stay at his village near Lahore. Before I could introduce myself, he began his rant about the ill-treatment he received from the crew in P. Intl. Airways when he asked for a drink and proceeded to asked me, the frequent flier with a history of 2 flights in 28yrs, of my informed opinion on Emirates. He was visibly pleased when I told him that I thought highly of Emirates and mumbled that the steward's and stewards were very friendly.

His next question didnt surprise me: Will drinks be served now? Do they serve liquer? Where was I from, India?. What do I think of the resistance in the U.S senate to the ongoing nuclear deal with India -- surprising me with his command over Indian current affairs. India and Pakistan are like brothers split apart, he proclaimed, and started addressing me as his "brother". He spoke of his 6-month long stay in native village where he committed the crime of owning a piece of land which almost cost him his life, the bloody family feuds, corrupt customs officials at the airport in Pakistan, and confided in me, his brother of 10-minutes, how sad he was when he left behind his child and wife crying at the airport. The mention of his child & wife brought liquer to his mind and he asked the inevitable question: When will drinks be served?. When told to be patient, he amused himself by pressing the buttons in front of his seat. His prayers were answered soon. The stewardess arrived with her drinks trolley and asked us, assuming her usual business-like sweet voice- what we gentlemen would like. He asked her in an equally sweet voice what her name was, played a guessing game on her nationality (singaporean,nobody guessed) and finally also asked her with all his munna-bhai charm to choose a liquer for him that she best liked. I guess, she would have handed the trolley to him that moment, but she refrained and gave him some brandy. Little did she realize what the aftermath of her indiscretion would be. Drink after drink followed. His stories became repetitive and tears rushed to his eyes as he told of his wife and his child. He also spoke of his Sikh friends in London and even invited me for a `davat` at their place.

I had never met a guy with Pakistani background but already I was seeing the traits of the archetypal Punjabi which extends across border and religion: Prone to fits of laughter and fits of sadness, and always threatening to hug the next person. Everytime he wanted a drink he would compose himself removed seat belts and sashayed to the stewardess and somehow managed to get more. When asked how he managed this, he told me this: the trick is not to slur in your speech and assume the smoothest tight-lipped brit accent while asking "May I have a drink, please". I still dont believe he managed to trick them this way.

By evening, no tightlippedness could prevent him from the slur. He turned completely silly and defied all attempts to get him back to his seat. A lady in the plane had her chappal stolen and he was later found wearing it and refused to give it back. The cabin crew became wary of him and arranged to forcibly get him back to his seat and tie him with his seat belt. They probably also informed police in Heathrow by then. Poor thing kept shouting "Ki hoi, Choudui ##@$$$&*&" and spent the rest of the time in a half-conscious state drooling like a dog on my lap while I pretended as if everything was alright.

Heathrow arrived. Everybody was asked to be seated and the captain announced -- Mr Latif, please show up at the door first. He didnt as he was still with his "Choudui" in his native village. The police arrived and asked him questions, he answered in Punjabi first and after some prodding and pushing he realized was in Heathrow, got his accent and senses back and chatted amiably with them and even asked jokingly --what did they think? that he was the o s a m a? That was the last I saw of this little 'o s a m a'.

a problem

2006-08-27T06:32:00.000-07:00

Heres an interesting linear algebra problem: Let F be a field and let V be a finite dimensional vector space over F. If α₁ , ... , α_m are finitely many vectors in V, each different from zero vector, prove that there is a linear functional f on V such that

f(α_i) ≠ 0 , i = 1 , .., m

My attempts at proving this lead to this problem: Can a union of finitely many sub spaces of dimension N-1 (hyperplanes) cover a vector space of dimension N.? Which seems an interesting problem in itself.

a passing thought

2006-08-25T20:40:00.000-07:00

Not long ago, I happened to visit Pune again for a day to settle an ancient but pressing matter. A tight schedule notwithstanding, I could squeeze in some time to visit one of my old haunts - Popular book house and walked out with a book which caught my attention-"i have landed" by s.j. gould( blissfully ignoring its pricey tag).

In the afternoon, back in the train, as people were settling down in their seats I began reading an essay from the book "no science without fantasy , no art without facts", which using Nabokov's example argues that a ``restless scholar'' whose work straddles different disciplines of knowledge can indeed enrich the fields and not become just a jack of all trades and master of none. Nabokov, we all know was literary giant of his times, but his interests in lepidoptery and his career as a "lab drudge" are less well known. Quoting nabokov, Gould writes----

...I work on my personal research .. a study of the classification of american 'blues' based on he structure of their genitalia(miniscule sculpteresque hooks, teeth, spurs etc visible only under microscope), while I sketch in with the aid of various marvelous devices, variants of the magic lantern..My work utterly exhausts me ... To immerse yourself in the wondrous crystralline world of the microscope, where silence reigns, circumscribed by its own horizon, a blindingly white arena--all this is so enticing that I cannot describe it.

Nabokov obsession with detail is dealt with at great length in the essay and Gould's unfettered admiration and bubbling enthusiasm is all too apparent beneath his own precisely crafted sentences though he tries to subdue it somewhat with pedantic words. I must say I got infected too.

Again, quoting Nabokov he writes-- Both the writer of fiction and the naturalist drew on a profound delight in precise comparitive observation. For Nabokov, a work of nature was a like work of art or rather it was a profound work of art, by the greatest of all living artists, evolution, and as much a joy to the mind and a challenge to the intellect as a Shakespear's sonnet. Hence it deserved to be studied like it, with never ending attention to detail and patience.

As I wonder about all this, an impulse prompted me to look outside the train. It had not started yet. A rat manoevered through the turd in the tracks. Somebody throws out a unused plastic bag with Sambar like liquid in it down the tracks. A scavenger notices this and rushes to it and grabs it before the a rat has it and starts sucking the Sambar from the plastic. The train begins to move, or rather wallow through the turd of the tracks. I used to think a train journey can bring the mind to sharp focus but this left me completely befuddled. The barrage of senses -- smells, sights and thoughts was overwhelming. As I was about to bid my Nabokovian delights goodbye, the greenery of the landscape engulfs my view. A person occupies a seat in front as for time and engages me in chit chat. But I never recovered.

sredni vashtar's origins

2006-01-15T07:33:00.000-08:00

The internet anagram server is a sophisticated tool for people wanting to indulge in sophisticated idling. Key in a name and have a laugh at the flattering and the nasty statements it makes on your him/her . There are plenty of examples that speak aptly of the person or the thing: Clint eastwood - Old west action. Times of india - "foetid simian" etc. Hoping to find something interesting in my name I keyed in mine at the server. The character crunching was pretty fast but the results were mostly sterile, producing stuff like: A SAKI HARMS NIRVANA RUN. But I managed to trivially deduce a flattering variation - SAKI MUNRA NIRVANA RASH. And yes i think it is flattering to be a "rash" of somebody that great despite the gaffe in spelling (munr{a}).

Saki Munro reminded me of his famous short story with a mysterious title -- "sredni vashtar". People have already speculated on its origins -- see wikipedia for instance. But they all seemed `idle' speculations compared to what I found at the anagram server : "Dr Hess' variant".

So who was this doctor Hess? Whoever that might be - I firmly believe that Saki did have this in mind when naming him. I write this not just with the authority of being the human variant of Saki Munro rash but with some solid research backing it: Saki himself prophesies this in his story. Google for "the doctor would be proved right" and read this splendid story.

--

icfp 2005

2005-10-05T06:44:00.000-07:00

Haskell rocks. Iam still learning the langugage and am yet to write a full-blown application in this language but I already have an unshakeable opinion that this is "the" language for rapid prototyping. Some years back (2002) when I first heard of this competetion I thought this would get me off the ground with my haskell fundaes. I didnt have a team and made very slow progress. Needless to say I didnt submit a thing. Three years hence and I am still learning the language. Not that the language is too hard or anything- I simply dont use it often. Competing in the icfp contest in haskell is impossible in the present circumstances. When I speak of Haskell here the respones were not encouraging: :Ohh Pascal, we had it in our school but why are you learning Pascal now?

This year I participated in the ICFP 2005 in the C language -- which everybody knows with a modestly (?) sized team of 6. Being the first time for us I never expected a position in the final results. We only were here to experience first-hand 72 hrs of mad-frenzied programming. ContestCrackers we called ourselves (uncool, you say but it shows some of the chutzpah we carrried).

Atul had already done a couple of icfps before, so we had a workhorse amongst us whom we could trust. Myself and Jatin(of the Enduro-3 fame) worked on the Cop programs. Navendu wrote the basic data structures and the skeleton code for everybody to use. Atul worked on the robber and tejinder and adrija on an alternative robber. We adopted 'C' which everybody knew. Atul went about programming the robber rather fast and kept the quality of his program very high. By Sunday afternoon he had a program ready and tested. Jatin and myself kept discussing the strategy for cop for 2 days. On the third day we implemented something but quality issues kept us from submitting a intelligent cop. We only managed to get a no-brainer cop working -- even this failed the initial rounds.

The couple of weeks later towards Friday I worked on the cop again and brought it to a stable enough state. The cops could now catch Atuls robber working in tandem. On Saturday when the twisted specs where announced, I was still tweaking my old cop. Jatin, Tejinder and Navendu and Adrija were kept busy by other issues. On Sunday, I had a look at the twisted specs tweaking it just so that it satisfied the new specs and added no new intelligence to the cop. So also with the robber which Atul got working independently at his office. The results are out and as expected we didnt do well in the first round. But the twist results are encouraging. The results show two submissions, each almost as good as the other. They only differed in that one cop program used a car and the other didnt. I dont know why I thought a cop-on-car would give us some-edge. Well, anyway.

coates' public lectures

2005-08-10T22:57:00.000-07:00

here and and here

Dont miss the proof that 1 is not a congruent number by Fermat. Not profound but beautiful. I struggled with the proof myself and gave up trying to find solutions for x^4 - y^4 = c^2 which turns out to have no solutions by a corollary of the above theorem. Incidentally, this also proves that x^4 + y^4 = z^4 has no solutions. Was it this proof that prompted those famous words?.

Roosting in my mosquito net

2005-07-16T02:42:00.000-07:00

For as long as I can remember, I have been trying to make peace with the mosquitoes in my bedroom. They abound here, aggressively foraging for blood, sucking any warm co2 exhaling animal on their random excursions. Not even my emaciated body is spared. Their sizes vary, but some of the large ones bear testimony to the fact that there is no dearth of blood in these parts. I wonder why the Malthusian theory doesnt hold for them. Their means of subsistence should now have been incapable of sustaining this mad proliferation!.

I never could not talk peace with these j e h a d i s. For a long time, my hapless body's only defence was the humble mosquito mat. Then I switched to the mosquito-repellant cream, but they adapted quickly and started sucking my buttocks or and other vulnerable areas. Incidentally, they use their long tongues(proboscis) to introduce some anaesthic substance leaving the skin numb temporarily and do the drilling and sucking. So usually I wouldnt realize their looting my blood until they make a clean job of it and leave. If the bite lasted long enough, my not so sharp reflexes would try swatting them, only to realize they escaped, leaving me crying in agony.

Truce was reached late last year when I was gifted a mosquito net by my ever so caring parents. The arrangment is quite clumsy and elaborate: two ends tied to an almirah handle and chair and the other two to window handles. The sides are then tucked beneath the mattress. A good part of my time is spent in preparing to sleep, tieing all the knots, switching on the night lamp, entering my burrow and making sure all sides are well-guarded- but all this is time well spent. The mosquitoes, otherwise willing to trade their lives for some blood now respect my space and leave me in peace. Above all they save me from pangs of guilt in having to snuff these defenseless creatures to premature death. With a life span of only a few weeks, should one risk death and taste ambrosia or die with an empty tummy feeling good about sparing somebody? Mosquitos chose the former and I would have done the same if I were in mosquito's shoes and leave self-abnegation for humans ( who incidentally, consider them irritants out on a blood-sucking spree ).

the proof (part 2)

2005-06-15T23:10:00.000-07:00

Last post we inched closer to the proof that

For prime p ,

( p-1)! + 1 = p^a ------(1)

is only possible for p <=5. We will consider four cases of primes p = 8n±1 and p = 8n+3 and p = 8n+5 to study how 2^r powers of p behave `mod` 2^f(p).

For p = 8n±1 we extract all powers of 2 from n and write

p = 2^x * (2*m+1) ± 1

p^2 = ( 2^(x+1) + 1 ) `mod` 2^(x+2)

p^2^r = (2^(x+r) + 1) `mod` 2^(x+r+1) where r > 0.

Now substitute y = x+r+1 in the above

p^2^(y-x-1) = (2^(y-1) + 1) `mod` 2^y

holds provided y > x+1

For p = 8*n + 5 and p = 8*n+3,

p^2 = (2^3 + 1 ) `mod` 2^4
p^2^r = ( 2^(2+r) + 1 ) `mod` 2^(3+r)

Now substitute y = 3+r,

 p^2^(y-3) = ( 2^(y-1) + 1 ) `mod` 2^y

and so

 p^2^(y-3) = ( 2^(y-1) + 1 ) `mod` 2^y

Lets gets back to the proof to where we had left it that is to get a lower bound on 's' such that

 p^s = 1 mod 2^f(p)

If we find a number 2^r such that p^2^r ≠ 1 mod 2^f(p) it would mean s > 2^r.

From what we saw above, if 2^y is some power of 2 that divides (p-1)!,
for p = 2^x*(2*m+1)± 1

 p^2^(y-x-1) = (2^(y-1) + 1) `mod` 2^y

and integer y > x+1, x >= 3

 s > 2^(y-x-1)

and since `a' is a multiple of s,

 a > 2^(y-x-1)

and so


p^2^(y-3) = (2^(y-1) + 1 ) `mod` 2^y

for p of the form 8n+3 or 8*n+5

s > 2^(y-3)

and since `a' is a multiple of s,

a > 2^(y-3)

From my previous post , y could be chosen

 y = (p - 1 - 2^n)/2 + (2^n -1 )

where 2^n is greatest 2 power less than or equal to (p-1)

for p = 2^x*(2*m+1) ± 1,

 a >= s > 2^( (p - 1 - 2^n)/2 + (2^n -1 ) - x -1 ) = 2^(2^(n-1) + (p-1)/2 - 2 - x )

and for p = 8n+3,8*n+5

 a >= s> 2^( (p - 1 - 2^n)/2 + (2^n -1 ) - 3 ) = 2^( 2^(n-1) + (p-1)/2 - 4 )

for p = 11 = 8*1 + 3 n = 3 ,

  a > 2 ^( 1 + 8 -1 -3) = 16

for p = 13 = 8*1 + 5 , n = 3

 a > 2 ^( 2 + 8 -1 -3) = 32

for p = 17 w = 4 , x = 4

a > 2^( 16 - 1 - 4 -1 ) > 2^10

In fact for all primes p >= 17 and p < 32 , a > 2^10 and likewise for primes p between 33 and p < 64 ,

 a > 2^(32 - 2 - 5 )= 2^25

and in general for primes

 p >= 2^x+1 and p >2^(x+1)

a > 2^(2^x-x-2)

So all we now need show is

 2^x > 2*x+3

for x = 4 it is obviously true.
if it true for some m.

 2^m > 2*m+3

we show it is true for m+1, as

 2^(m+1) = 2 * (2^m) > 4*m+6 = 2*(m+1) + 3 + (2*m+1) > 2*(m+1) + 3.

2^x > 2*x+3 is true for x >=4

then

2^x - x - 2 > x + 1

raising both sides to 2 power

2^(2^x-x-2) > 2^(x+1)

Now since p lies betwen 2^x + 1 and 2^(x+1)

a > 2^(2^x-x-2) > 2^(x+1) > p

which is too large for p^a = (p-1)!+1.

█

the proof (part 1)

2005-06-15T22:28:00.000-07:00

We still need a proof for case p > 7 to show that for prime p

(p-1)! + 1 = p^a ------(1)

is only possible for p <=5.

In my last post , we have an estimate greatest power of 2 dividing (p-1)!.

f(p) >= (p - 1 - 2^n)/2 + (2^n -1 )

We must also have for (1) to hold that

p^a = 1 `mod` 2^f(p)

Our efforts will be to try to get a lower bound on a and if that proves to be too large (> p) for (p-1)!+1 to be equal to p^a then we are done.

We also know from Euler that

p^2^(f(p)-1) = 1 `mod` 2^f(p) since EulerPhi(2^(f(p)) = 2^(f(p)-1)

The smallest power s on p that leaves 1 as remainder mod 2^f(p)
should divide 2^(f(p)-1). Which means s is a power of 2. Most importantly, s should divide 'a' as well. All we need to prove then is that s becomes too large for p > 7.

So why is 's' special?
If 2^r >= s,
p^2^r = 1 `mod` 2^(f(p))

and if 2^r < s then
p^2^r /= 1 `mod` 2^(f(p))

We need not find 's' to prove that 'a' > 's' becomes too large. We could just find a number 2^r less than 's' but is still large enough so that p^2^r becomes too large for (p-1)!+1. For that, we need to study some properties of 2^r powers of p mod 2^f(p).

the last problem redux

2005-06-15T06:16:00.000-07:00

The problem were trying to prove is if (p-1)!+1 is a power of p
for p > 7. We have already seen the proof for p=7 here

In the spirit of the proof for p = 7 , we'll show that a is greater
than p, but first we need to answer this question:

If 2^f(p) is the greatest 2 power that divides (p-1)! then

p^a = 1 `mod` 2^f(p)

but what is the value of f(p) ?

If 2^n is the greatest power of 2 less than or equal to (p-1) then

f(p) = floor(( p -1 )/2) + floor(( p -1 )/4) + floor(( p -1 )/8) + ... floor((p-1)/2^n)

We will satisfy ourselves with a lower bound on f(p)

since 2^(n+1) > p-1 >= 2^n

2 > (p-1)/2^n >= 1

=> floor((p-1) / 2^n) = 1

so f(p) >= 1 + 2 + 2^2.......2^(n-1) = (2^n-1 )

But wait, we are overlooking many 2 factors between 2^n and p-1.

which is atleast (p - 1 - 2^n)/2

and hence f(p) >= (p - 1 - 2^n)/2 + (2^n -1 ).

Thats enough for the proof which is to follow in the next post.

LHS in Wilson's theorem and prime powers.

2005-06-15T05:13:00.000-07:00

(p-1)! + 1 is not a power for p for prime p > 5.

Couple of weekends back , I was bored and was aching to get out of the drudge of bug fixing. I was going through the exercises in Niven's chapter on congruences looking for some problem that would make my day. I picked this one as it seemed hard but was not marked hard in the exercises. It took two weekends to bring this problem down. At this rate i am worried I would never would finish niven's book.

But first a proof for the case p = 7, which is simple

let (p-1)! +1 = p^a

We know that 6! is divisible by 16. The first step is the hardest but how it hides the vain attempts before this.

We must have
7^a = 1 mod 16

We also know that since 7^2 = 1 mod 16 and 2 is the least
such positive power, `a' must be a multiple of 2.

6! is divisible by 9

7^a = 1 mod 9
but 7^3 = 1 mod 9 ( 3 being the least such power)

a is a multiple of 2 and of 3
a is a multiple of the lcm (2 , 3) = 6

but makes 7^a too large to be equal to 6! + 1.

The next post will contain the proof for p > 7 .( not because of lack of margin space in this post :-)

2005-04-01T20:42:00.000-08:00

all dressed up and raring to go.

hanif kureishi's essay

2005-03-26T02:04:00.000-08:00

"To write is to be puzzled a second time by one's experience; it is also to savour it. In such reflection there is time to taste and engage with your own
life in its complexity."

from Hanif Kureishi's article Something given

This was enough provocation for me to change the title of my blog.!

Enduro-3

2005-03-22T21:04:00.000-08:00

On Feb 4, 2005, three of us -Brijesh, myself and Jatin hit the road on our bicycles heading for nowhere to compete for the race called - Enduro-3. Enduro-3 as the website will tell you is an adventure race meant to test endurance or more precisely, it is meant to test which will break first (if anything does) in 3 days of cycling(140Km) and trekking(70Km) carrying a backbreaking ~15kg load in extreme conditions in the wilderness - your body or your mind.

Myself and Brijesh were already veterans in this race having done this in 2004 with Kartik – a complete fiasco that ended the very first day of the race. As we set off on that chilly Feb morning in a tempo carrying us and the cycles for the flag off at Kataria school, we could not help but bring the yesteryear experiences into discussions.Is bar aisa nahi karna ,waisa nahi karna. The plans for the first day were well charted, but, we had no clue on how to go about the race on the second and the third day of the race apart from fixing the food rations.

We called ourselves the “Crash Test Dummies” (very proudly) --after the very amateurish nature of our team. But let the name not deceive you as we had enough in us to give the best in the race some competition.

The key was practice, knowing what to expect, traveling light while also keeping risks to a minimum. We got ourselves preparing for this race since December. The Enduro organizers -- the NEF initially announced that it would allow a team with atleast one girl member in the team. Brijesh went girl hunting but in vain. Towards the end of January , we had almost given up hope when they announced category for all-men teams. Jatin became an obvious choice.

Captain Nair(Brijesh) took charge of team and its overall preparations , got us together for the meetings and took pains to ensure that we were not missing anything for the race and that we had the best of the equipment for the race. I remember--Jatin's cycle was too heavy and captain loaned his spare aluminium thunder bike. I didn't have a haversack to carry stuff and we all headed to MG road braving midday heat to that shop - Peak (who apparently were flooded with orders from other Enduro aspirants). Thanks to him, our bags were loaded for a safe cycling journey out in the wilderness.

Our fitness was of chief concern to us. Earlier in Jan, I had contracted an infection and was going through heavy doses of antibiotics. Luckily,with Feb arriving I was feeling much better and more so as I started kicking the pedals.

Captain Nair,was worried he had put on too much weight and might go against him. But he did well to recover his fitness, chiefly by the two practice runs(one with me and other with Jatin ) to Sinhgad.

It was Jatin on whom we were counting the most for the race, having contested in regional level cycling races before and with his fantastic temperament, we felt we had a fair chance if we just followed his footsteps(literally&metaphorically) . Jatin also planned a food programme for the race which was a 2000 cal per day diet regime consisting of dry fruits, whey based MRP ( meal replacement powder for the uninitiated, which we grew to abhor with all our guts), whey protein loaded chocolates and 1.5kg per person glucose.

We had very few practice runs. I, for one, was kept away from my bicycle throughout Jan because of the illness -- though earlier in december Brijesh and I had cycled to Sinhgad and back with poor timing. Jatin and Captain did a pratice run to Sinhgad and back in quick time -- and were soaring in confidence. I had not practiced as much, but the recovery from the illness boosted my confidence enough for the race.

The race started amidst great fanfare and pomp with gssl-3plm represented by no less than two guys to cheers us at the flag off at Kataria school. The photographs will show Arif and Nitin and his wife Slesha prominently. Arif arrived with a placard advertising our team's presence to the world. The burden of responsibilty was weighing heavy on us. We had to perform! -- for Arif -- and for ourselves.

The first day as the Enduro rule book will tell you involved cycling for about 125 km and a 8km walk. The second day was all trekking (45 km) and some cycling(15km) and the third day was all trekking. We cycled first to NDA where race and time checks would first be done. We covered this initial distance fast enough without exhausting ourselves. We soon, lined up for the "Rifle shooting" event. Rajyavardhan Jatin Choudary managed to hit bull's eye consistently while the rest of us contented ourselves on sparing the imaginary bull of the lethal shots.

We had no time to lose and rushed immediately in full gear to the next TC at the peakcock bay. The roads in NDA are a feast for any rider. We cruised over the smooth roads feeling the acceleration as the pedals yielded easily to the kicking legs. At the peacock bay, we stepped into a boat and began rowing about a prescribed path in the Khadakvasla lake. Inexperience was telling on us , as we noticed with envy a team consisting of a girl rode past us easily here. We had head enough to realize the flaws of not rowing in tandem which we quickly switched to and profited.

We got off the boat in quick time, TC-ed (time checked) and started off on the next course towards next TC via "Splendour country club" . At this point , we were still in full steam and already ahead of some NDA students in the race. We cycled steadily for about 10km and just ahead of the next TC we stopped for a breather and decided to reload on our fuel--water and food. We had Whey-Protein loaded chocolates,figs and plenty of water to down them. As we ate, a pack of NDA guys who had been chasing us all this while managed to get past us. There was nothing to panic-- the fact that we kept these guys at our heels was feat enought for us.Having finished our food, we rushed to the next PC near Panshet dam.The heat was beginning to tell on us and other teams too. Riders on bike would offer us lift to neighbouring village which we politely refuse hiding temptation. With the ghats nearing , the ride was more roller coaster like - long never ending uphill rides and short downhill rides. We could no longer ride the bicycles uphill and had to drag it up. Some NDA guys in order not to lose pace at uphill roads would push with one hand the cycles of the female memebers in the team while managing their on cycles on the other hand!!. We had our own ordeals to deal with.Till this point we were more or less together and none of us had to wait for the other to catch up heat and the pain in the toenail from shoebite got me behind other members of the team. I was taking short laboured steps and stopping every once a while to keep the breathing normal. I caught up with the rest of them eventually as they sat down to take a breather. Here, another team captained by a brigadier caught up with us. Interestingly, there were girl members in the team. With competition arriving, we urged ourselves to perform harder and started off again uphill and raced past some members of the competing team. As we walked on, we noticed the brigadier panting hard and trying to catch some sleep on a boulder in the shade. --- which was heartwarming ( or cooling should I say) --See we werent the only ones feeling the heat!. Encouraged , we took time off again to unload the aching backs and breathe easy for while. A notch not far uphill from our resting point was the highest point from which it was all downhill . We rushed downhill in no time with the cycle speed I am sure exceeding 60kmph all the while taking the jolts on our tender bottoms.

We had left the other team far behind. This was about 3 pm in the after noon. We had cycled and walked hard and were losing steam. We took one final break to reload on fuel-- MRP and figs which we barely managed to get down the gullet. By 5 pm we had finished the cycling leg of the race for the day. The other team was nowhere in sight. The cycles were handed over to NEF guys and we began walking the 8km flat terrain . By this time the body was tired but pains had numbed. As we approached the end we noticed that the brigadier team that lagged behind us was inching closer and was to soon get past us. It was clear that our cycling was good but we needed to improve on trekking speeds. It was twilight 6.30 odd when we reached the final destination at Kumble. The resting place was out in the open near a school. We noticed that all the NDA teams had managed to arrive before us . We were the 8th team to arrive. Not bad at all for Crash test dummies , used only to clicking mice.

We dined on the MRP again and opened our sleeping bags and slept as if on fever out in the open. On another day this would have been the perfect romantic night to sleep under the stars but right then the body pains and feverishness could not get us into any appreciative mood. Jatin and brijesh had some trouble sleeping but I soon slept forgetting all pains in the world under the serene gaze of the stars.

By 5:30 in the morning we were woken up rudely and were arranged in ranks and files to start off trekking again. The night was over but nightmares had just begun for us. In no time all the teams that arrived last night got ahead of us. We trekked down a beautiful looking ghat by a stony path left by what used to be waterfalls in the monsoon. Captain now started to lag behind complaining of a pain in the knee but wore a brave face. My toenail didnt bother me much and I just kept jumping from one stone down to the next. We had a few misteps and falls and bruises but all minor.

Downhill, we were severely short of water and began searching for villagers for a source of water. A shy girl from a nearby hut offered us water and we unabashedly drank all she had in a small container. We still needed plenty of water.We werent sure of the path and wandered a while asking passers-by of the next village. Luckily, we caught sight of a PC( passage check) where the gentlemen were kind enough to offer us a bottle of water and also pointed us to a nearby borewell . We filled up all our bottles, got the glucose mixed in all of them and started off again. The next PC was not too far but the flat terrain however seemed to increase my toenail pain. I needed to fix this urgently before it could get worse. I first tried removing some sponge from the helmet and wrapping it around the toe, but it did not help. I finally cut open the toe part of the shoe to finally get some relief. We had lost precious time in this slow tramping and by the time we reached the next TC at Randhwe , it was 11:30 am. We still had ghats to climb and lot many miles to cover.

Uphill on the ghats , we could see a slow line of coloured midgets-in-red dragging themselves uphill. Brijesh's knee was getting unbearable now. We stopped after every 10min and take a breather. By 1pm we reached a village up in the ghats and stopped to have midday meal of MRP again. The chocalatey complan like taste of it was nauseating,but I gulped it all the same. We still had a mother-of-all-treacherous-ghats ahead to finish before the cycling leg started. We were joined by a couple of youngsters who would accompany us in this crossing. They call themselves the "sweep team" which is obliged to stick with the last of the teams still competing in the race. Their plight seemed more miserable than us. They did'nt have water or food since they arrived in the morning.We werent short on food supplies so would give them dry fruits and some water too. The climb was particulary exhausting in the midday sun, climbing, stopping everying once in a while with hardly a shade to hide from the sun. There are photographs Jatin took showing our plight here. But we kept together all the while and egged each other up. How Brijesh managed we dont know. He would refuse to get up at each resting point. We knew that our endurance was being tested to the maximum here. It was already 4:30 pm when we reached the next TC. The sweep team were relieved to leave us.

We now had to cycle to the next TC 15km away where the next session of trekking was to begin. We felt a lot relieved pedaling on the bikes and cycled hard stopping only twice at the steep roads through the ghats. But with the the sun setting, I was losing hope that we could cover next 30 km on foot.

At 6:30 pm we started the trekking and Jatin said he was all set, but I had my doubts. It was already dark when we started and a new sweep team now accompanied us.
My body and mind were suffereing from complete exhaustion and a thirst no amount of water could quench. I felt all too eager to sit down and rest for night. Brijesh walked limping because of the knee and my stomach refused anything in. All the while Jatin tramped along ahead of us with absolutely no mental or physical complaints. The power chocolate I last had in the evening was attempting to make its way out of my mouth. I controlled it vain and had to puke. Jatin offered to carry my bag for while but could not move easily with it in the night with a faint torch . In a short while, we were lost in the woods. Sweep team that accompanied us was trying to contact by a walkie-talkie. This gave us time to rethink , and I first broached the topic of quitting. We decided about postpone discussion until reaching the next village . In the meanwhile, the sweep team could not make any contact and tried in vain to search for a path in the dark leading to the faint flicker of the next TC visible in the horizon. We decided at about 10pm in the night to start a camp fire and sleep in woods. NEF guys offered us chappati with pickle – the delights of which was a revelation hidden from us by urban office going life. Food never tasted better.

At about 12am, as I was wrapped all cozy in the sleeping bag , a villager with a gun wandered over to us. The NEF guys waved money at him and asked him to get us jokers spending night in the middle of nowhere, to the next village which he agreed unconditionally. I reluctantly got up and decided to continue with the rest, cursing our luck. Humour and endurance was desperately needed to get us past these miseries. The puking had stopped by now but the hiccups had started. Despite all this, the our pace was brisk and we made it to the next village by 2am. I had already decided to quit and put this across to the team clearly. Brijesh quickly followed suit. The NEF guys tried to encourage to complete the race as only 10% of the race was left. But I was firm on my stand-- I had reached my limits and decided to quit. But the hero of the day-- Jatin decided he would continue.!!

We made enquiries about the next bus to Pune and slept there in a temple. Jatin,meanwhile, got himself lost in the woods but managed to reach the destination by 6:30am in the morning. He slept till 9:00 am and continued the race till 3:30.pm. It was plain, he had that extra bit the rest of us did’nt have. It is difficult to convey the exact feelings as we quit. I found it difficult to keep my attention away from my body which was crying for rest. I was glad I took the decision to quit as I eased myself into the sleeping bag but the back of my mind kept nagging why I did not put that extra effort. The debate still rages in my mind. But, there is always the next time to learn from our mistakes-no?.

grothendieck

2004-10-08T04:42:00.000-07:00

Last week I read about Grothendieck's life & times published in the web by A.M.S. It provides a fresh insight into his life and times esp. when mathematics bloomed in the IHES campus. There is enough material in there to get interested in his original ideas -- on schemes and topos. I hardly can wait to get glimpse of the beauty in those ideas.Anyway its a joy , reading thru those pages.

There is a great deal written about him and yet he seems a very mysterious man. Apart from his ideas on war the doc offers no real insight into his non-mathematical life. His early life esp and how he came to develop a taste for mathematics is not very clear. He is the kind of person who was meant to further the frontiers of mathematics --almost a device meant to churn lemmas and theorems day in and day out.He might have had a rather interesting personal life but mathematics has swept everything aside.

tagore and borges

2004-05-24T20:59:00.000-07:00

I was reading the old patagonian express yesterday and found Borges trashing Tagore and his (kitschy and corny) poems -"All he writes about are moons suns ". In defence, Theroux says that perhaps the English translations never could do justice to the bengali originals - to which Borges says that if anything a English translation could only be better.

It could be that Borges said this in jest or to please Theroux and his English readers but when I read similar opinions on Tagore by Nobokov it led me to believe that they may have a point. Arent there able Bengalis around to make Borges to recant( it may be too late as Borges may be in his grave).
Has someone tried a better translation so far?. Are cultural sensibilities of a Bengali so beyond comprehension of a foreigner that it could understood as anything but trash by a scholar like Borges when translated?.