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 …

r solutions. That the congruent number property for 1 and 2 is equivalent to the solvability of a single equation in positive rational numbers (y2 = x4 1 for 1 and y2 = x4 + 1 for 2)

Dec 22, 2025 · A congruent number can be defined as an integer that is equal to the area of a rational right triangle (Koblitz 1993). Numbers (a,x,y,z,t) such that {x^2+ay^2=z^2; x^2 …

If n is congruent, then multiplying n by the square of a whole number gives another congruent number. For example, since 5 is congruent, it follows that 20 = 4 ¢ 5 is congruent.

After introducing rank of elliptic curves, we conclude a natural number N is a congruent number if and only if the corresponding elliptic curve EN(Q) has non-zero rank.

Mar 1, 2012 · If n n is a positive integer, we say the integers a a and b b are congruent modulo n n, and write a ≡ b (mod n) a ≡ b (mod n), if they have the same remainder on division by n n.

Open Problem: Give a simple criterion to determine whether or not a number n is congruent. When n is congruent, give an e ective algorithm to nd a rational right triangle whose area is n.

CONGRUENT NUMBERS The Congruent Number problem is: Let nbe a positive integer. Is there a right triangle, with the lengths of all three of its sides being rational numbers, that has area …

The Congruent Number Problem Given a number n, is it congruent? Example: n = 5 is congruent.

Numbers are congruent when they have the same remainder after being divided by a chosen whole number (called the modulus).