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For example, 5 is a congruent number because it is the area of a (20/3, 3/2, 41/6) triangle. Similarly, 6 is a congruent number because it is the area of a (3,4,5) triangle. 3 and 4 are not congruent numbers.
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 ...
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]
The orange and green quadrilaterals are congruent; the blue one is not congruent to them. Congruence between the orange and green ones is established in that side BC corresponds to (in this case of congruence, equals in length) JK, CD corresponds to KL, DA corresponds to LI, and AB corresponds to IJ, while angle ∠C corresponds to (equals) angle ∠K, ∠D corresponds to ∠L, ∠A ...
A sequence of possible hypotenuse numbers for a primitive Pythagorean triple can be found at (sequence A008846 in the OEIS). The area (K = ab/2) is a congruent number [17] divisible by 6. In every Pythagorean triangle, the radius of the incircle and the radii of the three excircles are positive integers.
The prototypical example of a congruence relation is congruence modulo on the set of integers. For a given positive integer n {\displaystyle n} , two integers a {\displaystyle a} and b {\displaystyle b} are called congruent modulo n {\displaystyle n} , written
Clement's congruence-based theorem characterizes the twin primes pairs of the form (, +) through the following conditions: [()! +] ((+)), +P. A. Clement's original 1949 paper [2] provides a proof of this interesting elementary number theoretic criteria for twin primality based on Wilson's theorem.
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 ...