Unique Properties of 65

• 65 is the smallest number that becomes square if its reverse is either added to or subtracted from it
• 65 is the difference of fourth powers of the first two primes 65 = 3⁴ − 2⁴
• 65 is the only number which gives a prime square, on adding as well as subtracting its reverse from it (65 + 56 = 11², 65 − 56 = 3²)
• 65 is the smallest composite number of the form n² + 1, where n is even
• 65 is the magic constant of the 5 by 5 magic square
• 65 is the smallest hypotenuse of two different primitive Pythagorean triangles (33, 56, 65 and 16, 63, 65)
• 65 is the smallest hypotenuse of a primitive Pythagorean triangle whose 3 sides are composite integers
• 65 is the smallest number that is a sum of 2 distinct nonzero squares in more than one way
• In one of the first applications of a backtracking computer program Dana Scott in 1958 showed that a chessboard with a 2-by-2 hole in its center can be tiled with all twelve pentominoes in 65 ways

Common Properties of 65