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Can you name the following numbers?
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Aiwendil
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All answers are natural numbers (integers greater than zero). No answers are repeated.
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Property
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Notes
The only prime that is one less than a perfect square

The smallest square that is also the sum of two consecutive squares
= 5^2 = 3^2 + 4^2
The only number that is one more than a perfect square and one less than a perfect cube
25 = 5^2, 27 = 3^3
The smallest number of colours needed to colour any planar map with no adjacent regions the same colour

The only prime number that is the sum of four consecutive primes
= 2 + 3 + 5 + 7
The only even prime

The greatest number of faces a regular polyhedron can have

The largest number of knights that can be placed on an empty 8x8 chessboard such that none of them can capture any other

The smallest perfect number (i.e. whose divisors add up to itself)
= 1 + 2 + 3
The only number other than zero and one that is the square of the sum of its digits

The maximal number of perfect cubes needed to sum to any positive integer

The smallest number that is both a prime cube and the sum of prime squares
= 2^3 = 2^2 + 2^2
The only integer of the form x^y = y^x, with x not equal to y
= 4^2 = 2^4
The smallest positive integer whose reciprocal has a pattern of more than one repeating digit
1/7 = 0.142857142857142857 . . .
The smallest square that is also the sum of four consecutive cubes
= 1^3 + 2^3 + 3^3 + 4^3
The only number (other than zero) that is twice the sum of its digits

One of only two positive integers that cannot be written as the sum of 8 (or fewer) positive cubes (239 is the other)

The number of convex regular polyhedra
With 4, 6, 8, 12, and 20 sides
The smallest number that is both a prime cube and the sum of two distinct prime squares
= 5^3 = 2^2 + 11^2
The smallest number that is the product of three different primes
= 2 x 3 x 5
The answer to life, the universe, and everything, but not to this quiz.
= 2 x 3 x 5
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