Day31


2:38 pm, 4th May 2026
Okay, daily blogs every single day now.
the iitg launch didnt quiet work as expected, fuck that.
this summer is going to change my life, there is so much shit i want to fucking do, so much shit. So much I want to learn, so much I want to think about, so much I want to build.
My work is going to completely consume me now, I'm gonna do so much math, CS, build so much every single day. Every single day!!! yall are gonna see it on these blogs. watch this gng. Witness History.
rn tho, I have a Number Theory endsem (lol rmbr this course from day 23.
But anyways, life is so fucking cool man. im genuinely like a fucking math genius, like i learn stuff so quickly, my problem solving skills are so good, ive also built a bunch of really cool projects. im literally going to be THE WHIZ KID BILLIONAIRE OF THIS GENERATION
lets have fun then, havent done math in a long time, really excited for this.
I'll have to learn continued fractions and partitions... lesgooo. Apostol!!!
2:56 pm
ok so i have to learn continued fractions, quadratic reciprocacity and partitions. partitions i think imma do the combinatorial way, lets just have fun. im geuinely so excited for this summer man, we are going to have so much fun. by the end of these 75 days, imma be worth alteast a million. and more importnatly, by the end of thsi summer, ill have done a bunch of really really cool shit, and ill be really really good, and ill have worked really really hard, and been completely locked in. i got too many places to be, not gonna let anything affect me at all.
let's get started with number theory now tho:
starting off w continued fractions. wonderwall on loop.
3:12 pm
Took a small overview of what continued fractions are from wikipedia, found the proof/algo that every rational number can be represtned... now let's start reading burton's chapter.
3:39 pm
btw i realised that the burden of not letting things affect me is not on things, like i cant tell a person to not do something that will hurt me, its on me to not give a fuck about that person's actions. anyways, washed my face, back to work:
pic1
burton starting now.
3:41 pm
every rational number has unique represetnaiton in simple continued fraction: proof by indction:
just gotta prove this for $\frac{k}{n}$ where $k\lt n$ prove for all $(k,n)$ with $k\lt n \lt m$. for m, if you have $\frac{r}{m}$ obv, $a_0=0$ so now you need $\lt a_1,a_2....a_t \gt=\frac{m}{r}$ where <> is obv the coefficients of the fraction. just put $a_1=\lfloor \frac{m}{r} \rfloor$ and then the rest of the thing is of the form $\frac{k}{r}$ where $k \lt r$ and we're done by strong induction.
lmao my bloggin webapp is glitching cos the less than greater than symbols form tags in html so its not quiet rendering right. fuck this, lets js learn, ill blog later.
well this is not really unique, cos $[a_0,...a_n]=[a_0,...a_n-1,1] $
basically you can end at 1 or not.
4:05 pm
oof i was working, then i stopped for a while, i then realised, i can just not do that. i can just work all day lmao. literally without stopping at all man lesgo.
4:33 pm
ok lets get to it. dancin'
aint stopping at alllll
4:44 pm
simple continued fractions theory is done....
kth converegence thing is cool, you find the recurence for numerator and denominator of this. you can prove $p_kq_{k-1}-q_kp_{k-1}=(-1)^{k-1}$ and then the rest of all properties, like even convergents are increasing towards r and odd ones are decreasing towards r are obv. everyhting here is proved by induction. lets move onto infinite now.
lol this song dancin' is goated, im listening to the sped up version. i remember me and kuku had watched the man city documentary in lucknow, in that bedroom, we used to watch it w the AC blasting. wonderwall. man i love those memories. i lovee you kuku. i love mummy papa. i love the little kid w big dreams in lucknow. imma make all of you proud.
5:14 pm
ooof this is so interesting. there is so much math i still have to do, so motivating man. lets keep going....
i think i learnt everyhting there is to know about infinit contined fractions, the pell equations part if left, but this is so interesting. this summer imma do all this,
bascialy the two things you get are, every irrational number has unique represtnation, infact you can recurslivey write out the exact htings ezily, and then you get a cloesness quadratic bound, pretty cool, and then it tells oyu that contd. fraction bound is infact the best possible.
5:21 pm
lets move onto partitions.
5:38 pm
i mean just read what ferre's shape is. will derive anything i need in the exam hall lmao.
5:50 pm
i mean js read some more defintions of partitions, some generating functions. this is gonna be the fun aprt lmao, deriving every single thing on the spot haha. perfect. exctied. lemme js look at quadracti recirpoctivt from MONT adn then we going.
oooh hadnt done math in so long, this feels soooo fucking good.
5:54 pm
just looked at the defintion, (x/p) = 1 if QR, -1 otherwise
lets go then haha, this is going to be really really fun i think, woohooo.
3:30 am, 5th may 2026
Well, The exam was really cool. First let's discuss the exam, questions, my solutions etc.
So when I was walking in, I was feeling sooo emotional? like im not sure, I was feeling something, like crazy, wow this may really be the last time im doing this at IIT Delhi. very very surreal and excited for the future too. I walkd in the lecture hall w my headphones on, walked up to Amitabha Tripathi Sir, he said "i thought you had dropped out of iit" and i told him no sir not yet, i got the paper and went to sit on my seat.
Here Is the question paper:
pic2
So first off looking at this, I was like tf, this seems really tough lmao. anyways i went one by one, doing what i knew how and going from there. this is cool, this is what you need to do everytime in life, things are gonna seem difficult, js do what you can at that point and action solves everything anyways.
lemme js describe/sketch my solns for everything here:
1. The answer is $1+v_2(n)$
the way i did this is: first prove that for any odd p, the gcd=g (say) wont be divisble by g.
let $v_p(n)=k$ then $p \nmid \binom{2n}{p^k}$
The proof of this is relatively ez, js notice $v_p(2n-t)=v_p(t)$ for $t\le p^k$ and then note that since we're calculating $v_p$ we only care about divisible by p terms and we're done.
then all thats left is $v_2$ and the way i did this is, just use legendre formula for v_p in factorial; ie. for and odd x, look at $\binom{2n}{x}$
the v_2 here is: let $n=2^k * a$ with $a$ odd
$\sum_{t=1}^{\infty} \lfloor \frac{2n}{2^t} \rfloor - (\lfloor \frac{x}{2^t} \rfloor + \lfloor \frac{2n-x}{2^t} \rfloor )$
and then note that for t=1,2... k+1 ; the first term in floor is infact an integer, and since x is odd, the other two are not, hence that term is bigger than 1 and hence atleast 1. so you get that $v_2 \ge 1+k$ and then note $x=1$ implies $v_2 \le 1+k$ so youre done. cool problem.
2. really ez, just write terms with denominator p and p+d seperately and then note $(p-1)!d!(-1)^d = (p-1) \times (p+d -d)(p+d - (d-1)) \cdots (p+d-1) \equiv (p+d-1)! \ \mod (p+d)$
and then ezpz
3. i mean, first assume n carmichael, then squarefree follows ez, then pick a=g generator and the next part follows, the converse is ezpz too, just prove individually for each prime factor.
4. I mean i just used $\frac{1}{p} \binom{p}{k} \equiv \frac{(-1)^{k-1}}{k} \mod p$ and then its ez
5. lmao i messed up this problem, forgot that 4n^2 is also a square, i was thinking this fucking 4 is so annoying, wish it wasnt here then it would be ez haha. anyways did do some stuff like 4n^2+3 is 7mod 12, so you cant have only $\pm 1$ factors. then the deduction is ez, js put $n=n!$ but couldnt prove this.
6-7 didnt really try much... I mean another thing was, this paper is obv not doable in 2 hours, so after 1:40 mins or so, the TA came and asked do you guys want extra time, I said yea atleast an hour. there's only 4 ppl giving the exam, two ppl walked out in 2 hours, cos ig they werent able to do problems anyways, me and amit were left...
8. okay this was cool, i guess kind of. this is celarly continued fractions only, cos like thats the best approximation. my soln was pretty cool, first i found the contd. fraction for $\sqrt{2}$ is $[1;2,2,2\cdots]$ and this i jsut sort of guessed by writing $x^2+x=1+1+x$ lol
then using the recursion for $p_n$ and $q_n$ \(the convergent fractions\), you find both satisfy recurrence $p_n=2p_{n-1}+p_{n-2}$ then solve this characteristic eqn, using the initial terms and find exact formulars for $p_n$ and $q_n$ and then its js some cool algebra. damn fun.
9. lol ive seen this in walk through combo, this was the first problem i did, ill js attatch walk through combo photo, cos thats the exact bijection i wrote:
pic3
10. interesting solution,
first off ignore the all positive case, thats ez. atleast one is negative, and one is positive, let $z \le 0 \le x$... let $x+y=n$, then you compute $z=-(n-3)$, for the cubic eqn, replace $y=n-x$ and you get:
$x^3+(n-x)^3=3+(n-3)^3$
this gives us a quadratic in $n$ on expanding, namely:
$n^2(3-x) + n(x^2-9) + 8=0$
so this implies $x-3 \mid 8$ so $x-3=[1,2,4,8]$ which is just 4 cases. js go through them. and see only x=4 gives integer solution for $n$ in the quadratic.
writing this blog listening to hindi songsss, tu tu tu meriiiii
well anyways once this was over, talked to sir for a while, Amitabhi Tripathi sir is so good man. efoere i submitted the paper i was js looking around the hall, like surreal as fuck, this is really real man, this is really happening, im really going to drop out huh. im really going to achieve my dreams huh. wow lets fucking go man.
but yea, me and amit talked to sir for a while, he talked about my plan to dropout, talked about a few solutions of mine, he seemed impressed cos apart from me, no one had even 3 problems. and my solutions were pretty cool too.
then he dropped us back to hostel, in his car, we were js talking, about plans, what i wanna do, he droped off amit then it was js me and him, and we drove around for a while just talking. said you should prolly finish college, i js told him i really cannot do this, im gonna be off to bigger and better things in banglore pretty soon. im obv really fucking good at math and cs, one of the best itw, but i cant do this college shit. i need to go achieve my dreams.
he said one day "one day i'll say I used to know Aditya". I love this sir man. he gave me his number, told me to stay in touch and meet him soon. this was so fucking cool, loved it. really really good experience.
Whiel walking back to hostel, i felt so fucking inspired man, like literally I can't even tell you. Mony, this is genuinly life man, this is it right here bruh. This summer, I am going to do so much cool shit. Work every single moment man, every single fucking moment. this is all i fucking care about bruh. omg this is real, my dreams are right fucking here.
ok anyways, back to work: iitg launch kind of failed, ahhh fuck wtv. i think we're gonna launch at a couple other schools cos the product is genuinely nice, but we're gonna have to think of a genuine business idea we can get funded for soon. it's okay I know i'll find a way.
gonna post this blog, post it on ig and twitter, and then go to sleep. antoher fun day tmrw, daily blogs every single day man.
all the way up.