## 2 puzzles: Squares, Cubes & Prime Numbers

(1)   Sum of the squares of the digits of a prime is another prime number

For example,   11,   1^2 + 1^2 = 2 is prime

the prime number   23,   2^2 + 3^2 = 13
the prime number   41,   4^2 + 1^2 = 17
the prime number   61,   6^2 + 1^2 = 37
the prime number   83,   8^2 + 3^2 = 73
the prime number   101,   1^2 + 0^2 + 1^2 = 2
the prime number   113,   1^2 + 1^2 + 3^2 = 11

the prime number   317

3^2 + 1^2 + 7^2 = 59   is a prime number

Sum of the cubes of the digits of a prime is another prime number

>> 821, 823, 827, 829   are primes

8^3 + 2^3 + 1^3 = 521   is prime
8^3 + 2^3 + 3^3 = 547   is prime
8^3 + 2^3 + 7^3 = 863   is prime
8^3 + 2^3 + 9^3 = 1249   is prime

Find other examples.

(2)   When replacing each digit of a prime with
(a) its square
(b) its cube
(c) (a) & (b)

results in two new primes

For example, the prime number   983

9^2 = 81,   8^2 = 64   &   3^2 = 9
9^3 = 729,   8^3 = 512   &   3^3 = 27

The new numbers,   81649   is prime   &   72951227   is prime

Find other examples. 1. Paul says: