Prime 19 is the smallest prime...

[enzocreti]

19 (a twin) is the smallest prime such that between it and the closest square surrounding it (16) there is a prime 17.
Another example is 41...between 41 and 36 there is 37. (36 is the closest square surrounding 41, being 7^2 further away than 6^2).


89 is the first non twin prime with this property, any Others?

[wblipp]

Originally Posted by enzocreti

19 (a twin) is the smallest prime such that between it and the closest square surrounding it (16) there is a prime 17.
Another example is 41...between 41 and 36 there is 37. (36 is the closest square surrounding 41, being 7^2 further away than 6^2).


89 is the first non twin prime with this property, any Others?

The squares get sparse much faster than the primes, so once you get out of the very small numbers, there are going to many primes between adjacent squares. There are 23 primes between (100)^2 and (101)^2, still small numbers.

[Dr Sardonicus]

Originally Posted by enzocreti

19 (a twin) is the smallest prime such that between it and the closest square surrounding it (16) there is a prime 17.
Another example is 41...between 41 and 36 there is 37. (36 is the closest square surrounding 41, being 7^2 further away than 6^2).


89 is the first non twin prime with this property, any Others?

For each positive integer n, the interval closest to n^2 on the left is [n^2 - n + 1, n], and closest on the right is [n^2, n^2 + n].

For each n > 1, there is one largest prime p < n^2, and one smallest prime q > n^2.

So, for each n > 1, there is at most one prime closest to n^2 on the left and one closest on the right.

There could of course be duplications, in that the smallest prime q > n^2 could be the same as the largest prime < (n+1)^2. It is also possible as far as we know (though no examples are known, and nobody actually believes there are any) that there is an n for which there are no primes between n^2 and (n+1)^2.

It is also possible (though I know of no examples) that the largest square less than n^2 (for n > 1) is closer to (n-1)^2 than to n^2, or the least prime greater than n^2 is closer to (n+1)^2 than to n^2.

In any case, there are at most about 2*sqrt(X) primes less than X which are closest to some square. There are about X/log(X) primes less than X, so for all but an infinitesimal proportion of primes p, there will be some prime closer to the nearest square than p is.

The largest square for which there is a unique prime on the left closer to it than to the preceding square appears to be 11^2 = 121, the prime being p = 113. (The prime before 113 is 109, which is closer to 100 than to 121.) The largest square for which there is a unique prime on the right closer to it than to the next square appears to be 17^2 = 289, the prime q being 293. (The next prime after 293 is 307, which is closer to 18^2 than to 17^2.)