: Mathematics: congruum

mathematical analysis > calculus > sequence > integer sequence > congruum

geometry > algebraic geometry > Diophantine geometry > Diophantine equation > congruum

number > number theory > Diophantine geometry > Diophantine equation > congruum

... > elementary function > polynomial > polynomial equation > Diophantine equation > congruum

algebra > polynomial > polynomial equation > Diophantine equation > congruum

mathematical analysis > equation > polynomial equation > Diophantine equation > congruum

congruum

In number theory, a congruum (plural congrua) is the difference between successive square numbers in an arithmetic progression of three squares. That is, if ${\\displaystyle x^{2}}$ , ${\\displaystyle y^{2}}$ , and ${\\displaystyle z^{2}}$ (for integers ${\\displaystyle x}$ , ${\\displaystyle y}$ , and ${\\displaystyle z}$ ) are three square numbers that are equally spaced apart from each other, then the spacing between them, ${\\displaystyle z^{2}-y^{2}=y^{2}-x^{2}}$ , is called a congruum. The congruum problem is the problem of finding squares in arithmetic progression and their associated congrua. It can be formalized as a Diophantine equation: find integers ${\\displaystyle x}$ , ${\\displaystyle y}$ , and ${\\displaystyle z}$ such that ${\\displaystyle y^{2}-x^{2}=z^{2}-y^{2}.}$ When this equation is satisfied, both sides of the equation equal the congruum. Fibonacci solved the congruum problem by finding a parameterized formula for generating all congrua, together with their associated arithmetic progressions. According to this formula, each congruum is four times the area of a Pythagorean triangle. Congrua are also closely connected with congruent numbers: every congruum is a congruent number, and every congruent number is a congruum multiplied by the square of a rational number.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Congruum)

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