Legendre's conjecture states that, for each positive integer n, there is at least one prime between n^{2} and (n+1)^{2}. Even a cursory check shows that there are in fact more than one. Just how many more? Let us perform a hands-on investigation.
Here are two hypotheses (both are apparently true):
(A) Legendre's conjecture. For each integer n > 0, there is at least one
prime between n^{2} and (n+1)^{2}. (B) For each integer n > 0, there are at least [] primes between n^{2} and (n+1)^{2}. The expression [x] denotes the integer part of x. |
Consider statement (B). Note that it is stronger than Legendre's conjecture (A) – that is, (A) follows from (B).
On this page, we are going to perform a computational check of statement (B);
thus we will also check Legendre's conjecture (A) at the same time.
Note, however, that a direct computational check is not a proof.
We can only check a finite number of values, whereas the above statements involve an infinite number of values of n.
It seems extremely unlikely – but still remotely possible – that an abnormally big prime gap might invalidate
the above statements. For now, we do know that prime gaps that large are never observed up to 18-digit primes.
(We can say this with certainty because the largest gap
between any primes up to 18 digits is only 1442; this gap happens
between the primes 804212830686677669 and 804212830686679111.
The interval
Back to statement (B). This statement is suggested, in part, by the following observations:
(1) For integer m > 6776941, each interval (m^{4/3}, (m+1)^{4/3}) contains a prime
(generalized Legendre conjecture, case 4/3).
(2) For positive integers m and n, one can show that a single interval (n^{2}, (n+1)^{2}) contains at least [] intervals (m^{4/3}, (m+1)^{4/3}). (3) Combining (1) and (2), we can expect statement (B) to be valid for n large enough – namely, for integer |
However, the n > 0 part of statement (B) does not seem to follow from anything
(other than a direct computational check).
So it is quite surprising that (B) apparently does hold for all positive n.
The table below presents a direct computational check of statement (B).
You can save this page locally, together with primefactors.js,
and re-run the check for any values of n;
just modify the assigned start
and stop
values for continuous range verification;
for example, showVerification(start=200,stop=300)
.minPrimes
variable.
(Search the code for minPrimes =
minPrimes = 1
,
we would check the classical Legendre's conjecture.)
Now that we know there are at least that many primes in the interval
How many primes? n n^{2} < primes < (n+1)^{2} √n expected actual OK/fail
Post Scriptum: Here are some additional hypotheses – all of them appear to be true, too:
(C) For each integer n > 23, there are at least [1.5]
primes between n^{2} and (n+1)^{2}.
(D) For each integer n > 137, there are at least [2] primes between n^{2} and (n+1)^{2}. (E) For each integer n > 425, there are at least [3] primes between n^{2} and (n+1)^{2}. Here, again, [x] denotes the integer part of x. We leave checking statements (C), (D), and (E) to the interested reader. |
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