0
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1answer
35 views

Sophie Germain primes

Why did Germain come up with her Germain primes? I am intrigued to know why Sophie came across these primes. Do they have any applications?
9
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0answers
145 views

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor ...
9
votes
1answer
176 views

Ramanujan's personification of small positive integers

I dimly recall reading somewhere (perhaps in "The Man Who Knew Infinity"?) that Ramanujan associated personalities (perhaps it was mystical personalities, e.g. specific gods and goddesses?) with small ...
0
votes
4answers
386 views

What is the definition of a positive integer?

I am reading the book "The Number-System of Algebra (2nd edition)". At the starting of page-4 the author writes: A positive integer is a symbol for the number of things in a group of distinct ...
2
votes
0answers
31 views

Reference for Hilbert numbers

I've been studying a little bit of number theory, and during such studies I came across this interesting reference to Hilbert numbers, that is, numbers of the form $4n +1$. My question is a purely ...
8
votes
2answers
171 views

Are there examples that we know a number is a rational number but we do not know what is its numerator and denominator?

Are there examples that we know a number is a rational number but we do not know what is its numerator and denominator? In order to say clearly, this number should given by a certain formula, such as ...
13
votes
2answers
141 views

Identity of a Mathematician Mentioned in Euler

I and several others are in the process of translating one of Euler's papers from Latin to English, in particular the one that the Euler Archive lists as E36. In it Euler proves the Chinese Remainder ...
3
votes
0answers
51 views

Question about the first step in Mann's original proof of the Schnirelmann-Landau Conjecture

I was reading Henry Mann's proof for the Schnirelmann-Landau Conjecture from 1942 which can be found in JSTOR here Today, the Schnirelmann-Landau Conjecture is known as Mann's Theorem: $$d(C) \ge ...
4
votes
0answers
78 views

The number of solutions of $ax^4 - by^4 \equiv 1$ (mod $p$) for a prime of the form $p = 4n + 1$

Weil writes in his paper "The number of solutions of equations in finite fields" that Gauss finds the number of solutions of $ax^4 - by^4 \equiv 1$ (mod $p$) for a prime of the form $p = 4n + 1$ in ...
19
votes
1answer
441 views

Why did Gauss think the reciprocity law so important in number theory?

Gauss's Disquitiones Arithmeticae centers around the quadratic reciprocity law. It seems that he developed the genus theory of integral binary quadratic forms to find a natural proof of the quadratic ...
30
votes
2answers
6k views

Yitang Zhang: Prime Gaps

Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific. *EDIT*$^1$: Are there any experts here who ...
8
votes
4answers
506 views

Connections between number theory and abstract algebra.

I haven't taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat's little theorem, the law of quadratic ...
10
votes
2answers
122 views

Origin of well-ordering proof of uniqueness in the FToArithmetic

In the Appendix to Ivan Niven's book "Numbers: Rational and Irrational", he proves the Fundamental Theorem of Arithmetic (FToA) without using Euclid's Lemma that if a prime divides a product, then it ...
5
votes
1answer
267 views

A question on Paul Erdős's research on Egyptian fractions

A good day to everyone! I have a (somewhat) intriguing question regarding Paul Erdős's papers on Egyptian fractions (e.g., his 2nd paper during his mathematical career was about this topic). My ...
3
votes
1answer
203 views

Early history of lower bounds on the prime counting function

Let $\pi (x)$ be the number of prime numbers less than or equal to $x$. Euclid's proof of the infinitude of primes gives a horrible lower bound of the type $ \pi (x)>> \sqrt{\log{x}} $. ...
3
votes
1answer
209 views

History of the study of rational points on the circle

What is the first known instance of a mathematician parameterizing rational points on the unit circle by the slopes of rational lines going through a rational point on the circle?
4
votes
1answer
122 views

Nagura's paper--can we substitute for the original upper bound?

This question concerns two results about primes. The first is J. Nagura's 1952 result, that there is a prime on the interval $[x, (1+1/5)x] $ for $x> 2103,$ which depends on the result derived ...
17
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1answer
1k views

What was Ramanujan's solution?

The wikipedia entry on Ramanujan contains the following passage: One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a ...
4
votes
1answer
6k views

History of Mathematics: Sophie Germain and Fermat's Last Theorem

Sophie Germain's greatest contribution to mathematics was in number theory. She discovered a special case of Fermat's Last Theorem which we now call the Germain Theorem. Stated precisely: The ...
4
votes
0answers
229 views

Divisibility notation history

I'm writing a paper project for school about divisibility, so I'd like to include a bit of history about that subject. I'm mostly interested in notation of $|$ sign used in past, but everything else ...
18
votes
1answer
654 views

Did Leonardo of Pisa prove $n=4$ case of FLT?

Reputable on-line sources agree that Leonard 'Fibonacci' proved the nonexistence of positive-integer solutions to $c^4 - b^4 = a^2$ . Yet my change to Wikipedia to reflect this was reverted. I hope ...
12
votes
1answer
2k views

Proof of Euler's Theorem without abstract algebra?

Every proof I've seen of Euler's Theorem (that $\gcd(a,m) = 1 \implies a^{\phi(m)} \equiv 1 \pmod m$) involves the fact that the units of $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. ...