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The conjecture that Frey's elliptic curve was not modular. The conjecture was quickly proved by Ribet (Ribet's theorem) in 1986, and was an important step in the proof of ...

Suppose a,b in N, n=ab+1, and x_1, ..., x_n is a sequence of n real numbers. Then this sequence contains a monotonic increasing (decreasing) subsequence of a+1 terms or a ...

There exists an absolute constant C such that for any positive integer m, the discrepancy of any sequence {alpha_n} satisfies ...

For any two integers a and b, suppose d|ab. Then if d is relatively prime to a, then d divides b. This results appeared in Euclid's Elements, Book VII, Proposition 30. This ...

A special case of the Artin L-function for the polynomial x^2+1. It is given by L(s)=product_(p odd prime)1/(1-chi^-(p)p^(-s)), (1) where chi^-(p) = {1 for p=1 (mod 4); -1 ...

A factorization algorithm which works by expressing N as a quadratic form in two different ways. Then N=a^2+b^2=c^2+d^2, (1) so a^2-c^2=d^2-b^2 (2) (a-c)(a+c)=(d-b)(d+b). (3) ...

A periodic sequence such as {1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, ...} that is periodic from some point onwards.

A method which can be used to solve any quadratic congruence equation. This technique relies on the fact that solving x^2=b (mod p) is equivalent to finding a value y such ...

Also known as the difference of squares method. It was first used by Fermat and improved by Gauss. Gauss looked for integers x and y satisfying y^2=x^2-N (mod E) for various ...

Let p_i denote the ith prime, and write m=product_(i)p_i^(v_i). Then the exponent vector is v(m)=(v_1,v_2,...).