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:
Copyright
© 2011 Alexei Kourbatov, JavaScripter.net.