Primes Between Consecutive Cubes:
How many primes are there between n3 and (n+1)3?

Legendre's conjecture states that, for each positive integer n, there is at least one prime between n2 and (n+1)2. On this page, we will investigate a related question: How many primes are there between n3 and (n+1)3?

Here are two hypotheses – and both of them appear to be true:
(A) For each integer n > 0, there are at least four primes between n3 and (n+1)3.
(B) For each integer n > 0, there are at least 2n + 1 primes between n3 and (n+1)3.

Note that if the above statement (B) is true, then statement (A) is also true. Indeed, for n = 1 both statements are easy to check – and both are true, while for n ≥ 2 statement (A) follows from (B) because 2n + 1 > 4 for every n ≥ 2. Statement (B) is suggested by these observations:
(1) For integer m > 1051, each interval [m3/2, (m+1)3/2] contains a prime (generalized Legendre conjecture, case 3/2).
(2) For positive integers m and n, each interval [n3, (n+1)3] contains precisely 2n+1 intervals [m3/2, (m+1)3/2], for example:

  • the interval [13, 23] contains three intervals [13/2, 23/2], [23/2, 33/2], [33/2, 43/2];
  • the interval [23, 33] contains five intervals [43/2, 53/2], [53/2, 63/2], [63/2, 73/2], [73/2, 83/2], [83/2, 93/2];
  • the interval [333, 343] contains 67 intervals [10893/2, 10903/2],... [11553/2, 11563/2]; and so on.

    Combining (1) and (2), we see that, since 10513/2 < 10893/2 = 333, statement (B) is true for n ≥ 33 provided that (1) is true. But we already tested statement (1) and, based on the knowledge of maximum prime gaps, (1) holds true for large numbers (from m = 1052 and up to 18-digit primes). However, when m and n are small, statement (1) does not help us establish (B). Therefore, now it is of particular interest to test statement (B) directly for small n. The table below presents a computational check of statement (B) for a range of consecutive small cubes – and our computational experiment shows that (B) is apparently true. There are at least 2n + 1 primes between consecutive cubes n3 and (n+1)3. (We have to remember, though, that a computational check alone is not a proof.)

          n      n3   <    primes    <  (n+1)3  How many primes?    OK/fail 
                                               Expected:  Actual: