# Fermat’s Last Troll

The Urban Dictionary defines the word troll as:

Troll

1. (n.) Sometimes compared to the Japanese ‘Oni’, a troll is a supernatural creature of Scandinavian folklore, whose race was thought to have carried massive stones into the countryside. Lives in hills, mountains, caves, or under bridges. They are stupid, large, brutish, hairy, long-nosed, and bug-eyed, and may also have multiple heads or horns. Trolls love to eat people, especially small children.
2. (n.) An ugly little plastic doll with big hair.
3. (v.) To fish by dragging bait behind a moving motor-boat.
4. (v.) The act of posting a deliberately provocative message to a newsgroup or message board with the intention of causing maximum disruption and argument

Based on the final definition, I would like to propose that Pierre de Fermat pulled off the greatest troll in mathematical history (possibly unintentionally).

Problem II.8 in the 1621 edition of the Arithmetica of Diophantus. On the right is the margin that was too small to contain Fermat’s alleged proof of his “last theorem”.

Fermat was a French lawyer and mathematician who is remembered for things like discovering an original method of finding the greatest and the smallest ordinates of curved lines, and researching number theory, analytic geometry, probability and optics. Basically, he did a lot of math stuff; however, his most famous contribution to mathematics is referred to as Fermat’s Last Theorem or FLT. This marvelous contribution was written in the margin of a book. Fermat simply wrote:

“It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.”

An incredibly poignant theorem, but he just didn’t have space to write out it’s proof. Fermat later proved FLT for the specific cases n=4, but left his too-wide-for-the-margin proof unwritten.  For the next 358 years mathematicians worked night and day to figure out exactly what couldn’t fit in that little margin.

On Monday 19 September 1994, Andrew Wiles, an English mathematician with a childhood fascination with Fermat’s Last Theorem, finalized a correct and elegant proof for FLT. Almost 400 years of work had finally culminated in a seemingly magical proof. It was a national holiday and people ran from their houses yelling “FERMAT IS SOLVED!!! THE WORLD IS AT PEACE!!!”. I kid. That didn’t actually happen, but someone did write a poem about it:

Fermat’s Last Theorem Proven

By Marion Cohen

Fermat said the proof was too large
to fit in the right or left marg-.
True, back of the paper
might help, but he said, “I’m in charge.”

Now Wiles didn’t mind paper waste.
In fact, it was true to his taste
to use up whole reams
to realize his dreams
and he crossed out instead of erased.

Fermat was all snickers and smiles
as he smugly stayed clear of the aisles
and he thought “they’ll be glum
“but that proof will succumb
“though it’s going to take quite a-Wiles.”

Cohen takes the point of view that Fermat was conscious of his choice to never write down his proof. This raises the question – did Fermat actually have a feasible proof? If he didn’t, was he planning on upsetting the mathematical community for 400 years?  Let’s look a little deeper.

Fermat’s note first appeared in a collection of his edited works that was published by his son Samuel in 1670. This wasn’t the only time that Fermat claimed to have a proof, but didn’t have the time or paper or something to write it out. More than a century later, mathematicians like Euler had to reconstruct proofs for many of Fermat’s theorems.

One way to address his note is to claim that Fermat’s comment was simply a private note that he never intended to publish. Simply a little “Hey self, write this awesome, brilliant proof down later. You didn’t write it now because you had no paper. Love, me”. This wouldn’t really make sense. In his day and age it was pretty normal to write and publish commentaries on ancient works. The publishers intentionally left large margins in such works so that modern commentators could write their comments and show their contemporaries how much smarter they were than their forefathers. Basically, we can be pretty confident that Fermat was at least considering that his work would be published.

When I was researching this subject, I came to the realization that this story could have ended differently. Fermat didn’t publish a lot of his work. What if he had decided to publish his commentary on the Diophantus after all? What if he decided to publish his commentary after he had worked out the proof for n=4  and then decided that he didn’t have a proof for the theorem after all? What if he had edited out his comment? We wouldn’t have Fermat’s Last Theorem. He wouldn’t have sent mathematicians on a blind-leading-the-blind hunt in the forest. Wiles may have never had a reason to devote his life to proving FLT. Essentially, it was fortunate for the development of number theory that Fermat wasn’t prone to editing his papers. Or did he intentionally avoid editing so as to troll future generations — make them think he had all these brilliant ideas and proofs, but end up just making them work and argue down (what he thought) was a rabbit hole?

Sir Andrew Wiles. Image copyright C. J. Mozzochi, Princeton N.J.

Wiles’s proof involves mathematics that wasn’t  invented or discovered (that’s the topic of an entirely different blog post) until centuries after his death. The mathematical historian Howard Eves once said that “Fermat’s Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published.” Did Fermat use his version of an online forum to post his version of a troll? Or did he genuinely believe he had a solution that would have been proved incorrect? Or did he actually have a brilliant proof that doesn’t use any of Wiles’s modern math? We will never know, but I think the image of Fermat as the first comment troll is the most fun.

Sources

http://www.urbandictionary.com/define.php?term=troll

Poem: Discovering Patterns in Mathematics and Poetry By Marcia Birken, Anne Christine Coon

http://en.wikipedia.org/wiki/Fermat’s_Last_Theorem#Fermat.27s_conjecture

http://www.mathpages.com/home/kmath195/kmath195.htm

https://cs.uwaterloo.ca/~alopez-o/math-faq/node26.html

# Numbers Courtesy of Fermat and Mersenne

A portrait of Pierre de Fermat, lawyer and amateur mathematician. Image: Author and painter unknown, via Wikimedia Commons.

It is not often a person contributes to a field they do not even work in the way Pierre de Fermat has contributed to the field of mathematics. Born to a wealthy leather merchant, Fermat received a bachelor’s in civil law from the University of Orléans and went on to become a lawyer, while at the same time engraving his name into math history books, doing said math just for recreation. His importance in mathematics lead to many theorems named after him, as well as numbers. These numbers are known as Fermat numbers, which are positive integers, that take of the form Fn = 2(2n) + 1, when n is nonnegative and an integer. For example for F1, F1= 2(2)+1= 5. The first five Fermat numbers are 3, 5, 17, 257, and 65537, and these numbers continue to grow to incredibly large magnitudes. Fermat believed this form created an infinite number of prime numbers, which are known as Fermat primes.

Fermat numbers are occasionally written as 2n+1, but since when n is greater than zero and Fn prime, n must be a power of two, the form Fn = 2(2n) + 1 is the common form for Fermat numbers. One of the main problems with Fermat claiming all these numbers are prime is the fact that they soon become too large to calculate for even today’s computers, let alone a man with his pen and paper in the 17th century. Unfortunately for Fermat, by the time the 18th century rolled around, he was dead. In 1732, mathematician Leonhard Euler found that F5, which is 4,294,967,297, is actually divisible by 641, most likely figuring this out from having a large amount of time on his hands. While this showed that some Fermat numbers are not actually prime, excluding when n=0 in the form 2n+1, it does not discount the fact that the Fermat number equation could still make an infinite number of primes, since there are infinite amount of Fermat numbers. However as of now, the only Fermat primes that are known are F0 through F4.

Now initially I found the idea of an equation, the equation here being Fn = 2(2n) + 1, that finds only certain prime numbers, most of which are way too large to even be calculated even 400 years after the equation for them was created, the equivalent to a student doing extra credit when he has a 98% in his class. What I’m trying to say is, I found Fermat numbers pointless and to be the 17th century mathematician’s version of a braggadocio. However, I know nothing and Gauss managed to find a relationship between “Euclidean construction of regular polygons and Fermat primes,” where he showed a regular 17-gon could be constructed. It was also found a regular n-gon can be created if n is the product of any number of Fermat primes and the number 2. These regular n-gons take the property of being able to be constructed with a compass and straightedge. Who would have thought one of the greatest mathematicians to ever live could leave me feeling so inadequate, at least mathematically.

Similar to Fermat numbers are what are known Mersenne numbers, created by 17th century French mathematician and music theorist Marin Mersenne. Yet again a person being a jack of all trades, except instead of being a master of none they were a master of a few or at least of one. Mersenne numbers take of the form Mn = 2n -1, and Mersenne primes are numbers that take that form which are prime. Mersenne believed that for n<=257, Mn was prime for n= 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, and the rest are composite. While this belief turned out to incorrect, he still got the name for the primes. Just like Fermat primes, it is unknown whether there are an infinite number of Mersenne primes, but as of now 48 Mersenne primes are known, the largest being 257885161-1, which again makes me wonder how much time do some of these mathematicians have on their hands.

Mersenne numbers were originally studied because of their connection to perfect numbers, which are positive integers that are equal to the sum of their divisors. Euclid proved that if the number 2n-1 is prime, then 2n-1(2n-1) is a perfect number, which many years later led to Euler discovering that all even perfect numbers come in this form. Another interesting fact is that the ten largest known prime numbers are Mersenne numbers. I personally find number theory incredibly interesting, partly because I like numbers and partly because how mathematicians are able to come with these theorems and proofs baffle me. I ultimately wonder if they had any true goals when thinking about these primes, or if it was just for the pure fun and interest in it.

References:

http://en.wikipedia.org/wiki/Fermat_number

http://interact.sagemath.org/edu/2010/414/projects/tsang.pdf

http://mathworld.wolfram.com/FermatNumber.html

http://mathworld.wolfram.com/FermatPrime.html

https://primes.utm.edu/glossary/page.php?sort=FermatNumber

http://en.wikipedia.org/wiki/Mersenne_prime

http://en.wikipedia.org/wiki/Marin_Mersenne

http://en.wikipedia.org/wiki/Pierre_de_Fermat

http://mathworld.wolfram.com/MersennePrime.html

# Math Tricks and Fermat’s Little Theorem

So you think you’re a math whiz. You storm into parties armed with math’s most flamboyant tricks. You can recite the digits of π and e to 50 digits—whether in base 10 or 12. You can calculate squares with ease, since you’ve mastered the difference of squares x2 – y2 = (x + y)(x – y). In tackling 572, simply notice that 572 – 72 = (57 + 7)(57 – 7) = 64*50 = 3200. Adding 72 to both sides gives 572 = 3249.

Image by Hashir Milhan from
Wikimedia Commons under
Creative Commons.

You can also approximate square roots using the truncated Taylor series x ≈ c + (x – c2)/(2c) where c2 is the closest perfect square less than or equal to x. So √17 ≈ 4 + (17 – 16)/(2*4) = 4.125, whereas √17 = 4.123105 . . ..

But do you know what number theory is? It’s not taught in high school, and everyone’s repertoire of math tricks needs some number theory. Mastering modular arithmetic—the first step in number theory—will make you the life of the party. Calculating 83 mod 7 just means find the remainder after dividing by 7: 83 = 11*7 + 6, so 83 ≡ 6 mod 7. But it’s actually easier since 83 = 7*12 + (-1), so 83 ≡ -1 ≡ 6 mod 7. Modular arithmetic reveals the secrets of divisibility. Everybody knows the trick to see whether 3 divides a number; you just add the digits and check if 3 divides that number. But the reasoning is obvious when you write m = 10nan + 10n-1an-1 + . . . + 10a1 + a0 where the ai are the digits of m. Each 10k has a remainder of 1 modulo 3 so man + an-1 + . . . + a1 + a0 mod 3. Using this method generates tricks for other integers.

For instance, if 13 divides m, then 13 divides a0 – 3a1 – 4a2a3 + 3a4 + 4a5 + a6 – 3a7 – 4a+ . . . and the pattern continues. This is because

10 ≡ -3 mod 13,

102 ≡ 10*10 ≡ (-3)(-3) ≡ 9 ≡ -4 mod 13,

103 ≡ 102*10 ≡ (-4)(-3) ≡ 12 ≡ -1 mod 13,

104 ≡ 103*10 ≡ (-1)*(-3) ≡ 3 mod 13,

105 ≡ 104*10 ≡ 3*(-3) ≡ -9 ≡ 4 mod 13,

and 106 ≡ 105*10 ≡ 4*(-3) ≡ -12 ≡ 1 mod 13.

From 106 and onwards the pattern repeats. In fact, calculating 10n mod k for successive n will reveal the divisibility rule for k.

Then comes Fermat’s little theorem, the key to solving seemingly impossible calculations.

Fermat. Image from Wikimedia
Commons. Under public domain.

The theorem states for a prime p and integer a that aa mod p. If p doesn’t divide a, then  ap -1 ≡ 1 mod p. I’ll illustrate the power of this little result in a computation. Let’s find 2371 mod 5. We’ll be using 24 ≡ 1 mod 5, which we get from Fermat’s little theorem. Now 2371 = 236823 =(24)9223, so by the theorem, 2371 = (24)9223 ≡ 19223 ≡ 1*23 ≡ 8 ≡ 3 mod 5. Exploiting Fermat’s little theorem can impress your friends, but try to avoid questions. Computing residues modulo a composite number—calculating b mod n for a composite number n—may require paper and ruin the magic.

Leonhard Euler proved a more general version of Fermat’s little theorem; it’s called the Euler-Fermat theorem. This theorem isn’t for parties; explaining it to the non-mathematically inclined will always require paper and some time. Nonetheless, it will impress at dinner if you have a napkin and pen.

Understanding this theorem requires Euler’s totient function φ(n).

Euler. From Wikimedia
Commons. Under public domain.

The number φ(n) for some n is the number of positive integers coprime with n that are less than or equal to n. Two numbers a and b are coprime if their greatest common factor is one. Hence 14 and 3 are coprime because their biggest shared factor is 1, but 21 and 14 aren’t coprime because they have a common divisor of 7. Moreover, φ(14) = 6 because 14 has six numbers less than or equal to it that are coprime with it: 1, 3, 5, 9, 11, and 13. Notice that if p is prime, φ(p) = p – 1 because every number less than p is coprime with p.

Now the Euler-Fermat theorem states that aφ(n) ≡ 1 mod n, which looks similar to ap -1 ≡ 1 mod p for a prime p. In fact, if = φ(p) = p – 1 for a prime p, the theorem reduces to Fermat’s little theorem.

Fermat’s little theorem has another generalization, Lagrange’s theorem. Joseph-Louis Lagrange was Euler’s student. Lagrange’s theorem generalizes both the previous theorems and doesn’t even require numbers. But due to the required background in group theory, I won’t go over the theorem. You can find links to more information on Lagrange’s theorem below.

Remember, a math whiz doesn’t need props like a magician does. Hook your audience with some modular arithmetic, and reel the people in with Fermat’s little theorem. If you want to get complicated, the most you’ll need is a pen and some paper.

Sources and cool stuff:

Modular arithmetic: http://nrich.maths.org/4350

Proof of Fermat’s little theorem: https://primes.utm.edu/notes/proofs/FermatsLittleTheorem.html

Euler-Fermat theorem and its proof: http://www.artofproblemsolving.com/Wiki/index.php/Euler%27s_Totient_Theorem

Lagrange’s theorem (only for the brave): http://cims.nyu.edu/~kiryl/teaching/aa/les102903.pdf

Number theory textbooks: Gordan Savin’s Numbers, Groups, and Cryptography and George E. Andrews’s Number Theory

Interesting sources of math tricks and problems: Paul Zeitz’s The Art, Craft of Problem Solving and The USSR Olympiad Problem Book, and What is Mathematics by Richard Courant and Herbert Robbins

# Leonhard Euler: Euler’s Identity, Fermat’s Last Theorem, and the Basel Problem

Leonard Euler. Image: public domain, via Wikimedia Commons.

Leonhard Euler was born in 1707 in Basel, Switzerland, and died in 1783. Over the course of his life he published many articles, some related to fields such as physics and astronomy, but many in the field of mathematics. In mathematics, three contributions he made are Euler’s Identity, his work on Fermat’s Last Theorem, and his solution to the Basel problem. Euler’s Identity is part of the mathematical field of Complex Analysis, which involves the application of various mathematical concepts for complex numbers a + bi, where i = √-1. Fermat’s Last Theorem is part of number theory, a field focused on the relationships between numbers, but primarily the integers. The Basel Problem involves summations, an important part of Calculus where it is used to solve difficult integrals, as well as other areas of Mathematics.

One of Euler’s more profound equations is Euler’s Identity, which states that e +1 = 0. This is a special case of Euler’s formula, which shows that eix = cos(x) + isin(x). Therefore, in the case of Euler’s Identity, x = Π, so e = cos(Π) + isin(Π) = -1 + i(0) = -1. By adding 1 to both sides, the standard result of e +1 = 0 becomes apparent.

This Identity has been considered by many to be “remarkable” and “among the most beautiful formulas in mathematics” for a variety of reasons. First, it includes both the number one and the number zero, the multiplicative and additive identities. Furthermore, it involves the constant Π, the most fundamental constant in geometry, and e, also known as Euler’s number, which is the base of the natural logarithm and can be used in many real world applications. Finally, there is i, the fundamental imaginary unit equal to the square root of negative one.

Euler also worked on Fermat’s Last Theorem. The theorem states that for the equation xn + yn = zn, there are no non-zero integer solutions for n equal to any value greater than two. Euler, in a letter to his friend Christian Goldbach, claimed to have a proof of the theorem for the case where n is equal to three. However, part of his proof was incorrect. He relied upon proving that for any numbers p and q where p2 + 3q2 is a cube, there exist numbers a and b where p2 + 3q2 = (a2 + 3b2)3. He incorrectly attempted to prove this using imaginary numbers, and as such invalidated his proof. Despite this, it could still be argued that he proved the case where n equals three as other mathematicians have used some of his other works to correct his mistakes. Additionally, Euler also proved the case of Fermat’s Last Theorem where n is equal to four.

The Basel problem is finding n=11n2. Pietro Mengoli, an Italian priest, originally constructed the problem in 1644; in later years, both Jakob Bernoulli and Gottfried Leibniz, two prominent mathematicians of the time, were unable to solve it. This led to the Basel problem becoming a sort of challenge for mathematicians at the time. It is unknown when Euler first began work on the Basel problem; however, in 1731, he calculated an approximation of the first one thousand terms 112 + 122 + 132 + … + 110002. However, we have learned that his approximation of 1.64393 is only accurate to the first two decimal places. Euler continued to develop more and more accurate approximations, eventually ending up with the approximation 1.644934, which is accurate to six decimal places.

In 1734, Euler published his first proof that the Basel problem was in fact equal to Π26, which he had previously noticed was approximately equal to 1.644934, his most accurate approximation for the Basel problem. However, he made many logical pitfalls in this intial proof, lending it to the criticism of Daniel Bernoulli, a mathematician and one of his contemporaries. In 1741, Euler had completed his final proof, which remedied the problems of the first. I will now show how Euler proved this, using a result from an earlier proof that Π28 = 1 + 19 + 125 + 149 + …, which is required for this proof to be valid.

First, Euler expanded the sum and split it into fractions with even and odd denominators n=11n2 = (1 + 19 + 125 + 149 + … ) + (14 + 116 + 136 + 164 + …). He then observed that the sum of fractions with even denominators is equal to one forth of the original sum (1 + 19 + 125 + 149 + … ) + (14 + 116 + 136 + 164 + …) = (1 + 19 + 125 + 149 + …) + (14)(1 + 14 + 19 + 116 + …). He already knew that the sum of the fractions with odd denomitors was Π28. He then substitiued it and the original sum (1 + 19 + 125 + 149 + …) + (14)(1 + 14 + 19 + 116 + …) = Π28 + 14n=11n2. Euler then subtracted one fourth of the original sum from both sides of the equation, and then multiplied both sides by four thirds to get the final value of Π26. Π28 + 14n=11n2 –> (34)∑n=11n2 = Π28 –> ∑n=11n2 = Π26

The Basel problem is a special case of the p-series n=11np where p = 2. While we now know that the series diverges for p less than or equal to 1 and converges for p greater than 1. Euler worked on the p = 3 case, but his brute force approximations (as initially used in the Basel problem) did not yield any values he recognized. In 1978, it was proven that the number the p=3 case converges to is irrational, but nothing is known about the odd values of p greater than 3.

These are but three of Leonhard Euler’s many contributions in the field of Mathematics. I wrote about these three because I find all of them to be very interesting. Euler’s Identity, of course, has the many mathematical constants, and it amazes me how so many can be related in one equation. Fermat’s Last Theorem is perhaps the most infamous problem in Number Theory, as Fermat scribbled it in the margin of a journal, claiming to have an incredible proof but it would not fit in the margin. And The Basel Problem is the only p-series where the exact sum of the series is known.

Sources:

http://www.amt.edu.au/euler.html

http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/

http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat’s_last_theorem.html

http://www.biography.com/people/leonhard-euler-21342391#early-life-and-education

# Marveling at the Indian Genius: Ramanujan

Hardy left the cab. He was visiting an ill Ramanujan at Putney. Ramanujan asked about his trip. Hardy remarked the ride was dull; even the taxi number, 1729, was dull to the number theorist. He hoped the number’s dullness wasn’t an omen predicting Ramanujan’s declining health. “No Hardy,” Ramanujan replied, “it is a very interesting number.” The integer 1729 is actually the smallest number expressible as the sum of two positive cubes in two different ways. Indeed 1729 = 13 + 123 = 93 + 103.

Ramanujan. Picture courtesy of Konrad Jacobs. From Wikimedia Commons

The mathematicians Srinivasa Iyengar Ramanujan and G.H. Hardy comprise the characters of this story. Ramanujan’s casual discovery of the smallest so-called taxicab number was no fluke. Ramanujan and Hardy’s most famous result was an asymptotic formula for the number of partitions of a positive integer. A partition of a number n is a way of writing n as the sum of positive integers. Reordering terms doesn’t change a partition. Thus the partitions of 5 are 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1; hence 5 has 7 seven partitions. Counting the partitions becomes difficult as n grows. Ramanujan had made several conjectures based on numerical evidence, and Hardy credits many of the needed insights to Ramanujan. The formula is complicated and counter-intuitive. It involves values of √3, π, and e, all very strange numbers for a counting formula. Near his death, Ramanujan discovered mock theta functions, which mathematicians are still “rediscovering” today.

These are only the achievements of later Ramanujan. The early life of Ramanujan is even more surprising. He spent his childhood in the poor south Indian town, Kumbakonam. In school, he scored top marks. He probed college students for mathematical knowledge by age 11. When he was 13, he comprehended S.L. Loney’s advanced trigonometry book. He could solve cubic and quartic equations—the latter method he found himself—and finished math tests in half the allotted time. If asked, he could recite the digits of π and e to any number of digits. He borrowed G.S. Carr’s A Synopsis of Elementary Results in Pure and Applied Mathematics by age 16 and worked through its many theorems. By 17, he independently investigated the Bernoulli numbers, a set of numbers intimately connected to number theory.

So How did Ramanujan do it? No brilliant teacher—excluding Hardy and J.E. Littlewood when Ramanujan was already an adult—taught Ramanujan. The odds were against Ramanujan from the start. Ramanujan often used slate instead of paper because paper was expensive; he even erased slate with his elbows, since finding a rag would take too long. He lost his first college scholarship by neglecting his every subject that wasn’t math. The first two English mathematicians he sent letters to request publication did not respond; was it luck that Hardy did?

Ramanujan credits Namagiri, a family deity, for his mathematics. He claimed the goddess would write mathematics on his tongue, that dreams and visions would reveal the secrets of math to him.

Ramanujan was undoubtedly religious. Before leaving India, he respected all the holy customs of his caste: he shaved his forehead, tied his hair into a knot, wore a red U with a white slash on his forehead, and refused to eat any meat. In the Sarangapani temple, Ramanujan would work on his math in his tattered notebook.

Hardy, the ardent atheist, didn’t believe that gods communicated with Ramanujan. He thought flashes of insight were just more common in Ramanujan than in most other mathematicians. Sure, Ramanujan had his religious quirks, but they were just quirks.

Looking at Ramanujan’s work, I find it hard to deny that some god aided the man. I cannot even imagine Ramanujan’s impact had he lived longer or had better teachers.

Sources and cool stuff:

biographical stuff: http://www-history.mcs.st-andrews.ac.uk/Biographies/Ramanujan.html, Christopher Syke’s documentary Letter’s from and Indian Clerk, Robert Kanigel’s The Man Who Knew Infinity

Ramanujan and Hardy’s original paper on partitions: http://plms.oxfordjournals.org/content/s2-17/1/75.full.pdf

Wolfram on taxicab numbers: http://mathworld.wolfram.com/TaxicabNumber.html

Wolfram on Hardy-Ramanujan number: http://mathworld.wolfram.com/Hardy-RamanujanNumber.html

Wolfram on Bernoulli numbers: http://mathworld.wolfram.com/BernoulliNumber.html

Ramanujan’s papers and notebooks: http://www.imsc.res.in/~rao/ramanujan/contentindex.html

mock theta functions: http://www.ams.org/notices/201011/rtx101101410p.pdf