# 125

This number is a composite.

125 = (10^{2} + 5^{2}) or (11^{2} + 2^{2}), i.e., the sum of prime squares in two ways. [Sladcik]

125 = 5^{3} (a prime cube). It is the smallest
number which is both a sum of distinct prime squares and a prime cube. [Kulsha]

Conjecture by Zhi-Wei Sun to be the largest power
(5^{3}) for which there is no prime between it and
the previous power (11^{2}). The other prime gaps
between powers are in (2^{3}, 3^{2}),
(2^{5}, 6^{2}) and
(5^{2}, 3^{3}). [Caldwell]

The smallest prime cube (p^3), where (p+3) = the digit-sum of (p^3) = another prime cube. Note that 125 = π(12)^π(5). [Worrom]

125 = sigma(prime(1*25)) + prime(1) + 25. [Worrom]

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