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Triangle with the area 6, a congruent number. In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. [1] [2] A more general definition includes all positive rational numbers with this property. [3] The sequence of (integer) congruent numbers starts with
There are other prime-related congruences that provide necessary and sufficient conditions on the primality of certain subsequences of the natural numbers. Many of these alternate statements characterizing primality are related to Wilson's theorem , or are restatements of this classical result given in terms of other special variants of ...
A prime number with this property is sometimes called a Mirimanoff prime, in analogy to a Wieferich prime which is a prime such that 2 p-1 ≡ 1 (mod p 2). The existence of primes satisfying such congruences was recognized long before their implications for the first case of Fermat's Last Theorem became apparent; but while the discovery of the ...
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions.German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."
Tunnell's theorem states that supposing n is a congruent number, if n is odd then 2A n = B n and if n is even then 2C n = D n. Conversely, if the Birch and Swinnerton-Dyer conjecture holds true for elliptic curves of the form y 2 = x 3 − n 2 x {\displaystyle y^{2}=x^{3}-n^{2}x} , these equalities are sufficient to conclude that n is a ...
If a is any number coprime to n then a is in one of these residue classes, and its powers a, a 2, ... , a k modulo n form a subgroup of the group of residue classes, with a k ≡ 1 (mod n). Lagrange's theorem says k must divide φ ( n ) , i.e. there is an integer M such that kM = φ ( n ) .
Pages in category "Theorems in number theory" The following 106 pages are in this category, out of 106 total. ... Ramanujan's congruences; Ribet's theorem;
Montgomery reduction; Modular exponentiation; Linear congruence theorem; Method of successive substitution; Chinese remainder theorem; Fermat's little theorem