Department of Mathematics

The film "Cube" and prime numbers

There are 193 numbers below 1000 which are a prime or a power of a prime. This total is made up of the 168 primes less than 1000, namely

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997

plus the 11 numbers which are squares of primes

4 = 2², 9 = 3², 25 = 5², 49 = 7², 121 = 11², 169 = 13², 289 = 17², 361 = 19², 529 = 23², 841 = 29², 961 = 31²

the 4 numbers which are cubes of primes

8 = 2³, 27 = 3³, 125 = 5³, 343 = 7³

the 3 numbers which are fourth powers of primes

16 = 2⁴, 81 = 3⁴, 625 = 5⁴

and the following seven numbers which are higher powers of primes

32 = 2⁵, 243 = 3⁵, 64 = 2⁶, 729 = 3⁶, 128 = 2⁷, 256 = 2⁸, 512 = 2⁹

So the probability that a single number less than 1000 chosen at random is not a prime or a power of a prime is approximately (999-193)/999 = 0.807, and the probability that three chosen at random all have this property is approximately 0.807³ = 0.525, or slightly better than 50:50.