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if p ≡ 3 (mod 8), then p is not a congruent number, but 2 p is a congruent number. if p ≡ 5 (mod 8), then p is a congruent number. if p ≡ 7 (mod 8), then p and 2 p are congruent numbers. It is also known that in each of the congruence classes 5, 6, 7 (mod 8), for any given k there are infinitely many square-free congruent numbers with k ...
A congruent number is defined as the area of a right triangle with rational sides. Because every congruum can be obtained (using the parameterized solution) as the area of a Pythagorean triangle, it follows that every congruum is congruent. Every congruent number is a congruum multiplied by the square of a rational number. [7]
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 ...
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 ...
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, then () is congruent to modulo n, where denotes Euler's totient function; that is
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d.
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. [1]
For two polyhedra with the same combinatorial type (that is, the same number E of edges, the same number of faces, and the same number of sides on corresponding faces), there exists a set of E measurements that can establish whether or not the polyhedra are congruent. [7] [8] The number is tight, meaning that less than E measurements are not ...