Multinumbers

Multinumbers are natural numbers that can be expressed as a sum of two squares in at least two distinct ways. Formally, a number n is called a multinumber if there exist two distinct unordered pairs {a, b} and {c, d} of positive integers such that a^2 + b^2 = c^2 + d^2 = n, with the pairs differing up to permutation. Representations that only differ by sign or by swapping the order of a and b are not counted as distinct.

The term is informal and not part of a universally adopted mathematical taxonomy. It is used in popular expositions and puzzle contexts to highlight numbers that admit multiple representations under a fixed additive form. The concept can be extended to other representations, such as sums of three squares or sums of higher powers, though the available theory varies with the form considered. In the two-squares case, the number of representations relates to classical results like Fermat’s theorem on sums of two squares and associated divisor counts, which determine when a number is expressible as a sum of two squares and, in some cases, how many distinct representations it has.

Examples include 50, which equals 1^2 + 7^2 and 5^2 + 5^2; 65, which equals 1^2 + 8^2 and 4^2 + 7^2; 85, which equals 2^2 + 9^2 and 6^2 + 7^2; and 325, which equals 1^2 + 18^2 and 10^2 + 15^2.

See also sums of two squares, Fermat’s two-squares theorem, and taxicab numbers.