A229488 Conjecturally, possible differences between prime(k)^2 and the previous prime for some k.

1, 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 38, 42, 44, 48, 50, 54, 56, 60, 62, 66, 68, 72, 74, 78, 80, 84, 86, 90, 92, 96, 98, 102, 104, 108, 110, 114, 116, 120, 122, 126, 128, 132, 134, 138, 140, 146, 150, 152, 156, 158, 162, 164, 168, 170, 174, 176, 180

Offset: 1

Author

T. D. Noe, Oct 21 2013

Keywords

• nonn

Comments

Are there any missing terms? The first 10^7 primes were examined. All these differences occur for some k < 10^5. Note that the first differences of these terms is 1, 2, 4, or 6.

From R. J. Mathar, Oct 29 2013: (Start)

This sequence of possible differences d= prime(k)^2 -q looks similar to A047238; 1 is an exception associated with the single even prime, 1=2^2-3.

[Reason: Otherwise primes are odd, squared primes are also odd, so the differences are even and therefore in the class {0,2,4} mod 6.

Furthermore primes are of the form 3n+1 or 3n+2, squared primes are of the form 9n^2+6n+1 or 9n^2+12n+4, so squared primes are of the form ==1 (mod 3).

The difference prime(k)^2-q is therefore the difference between a number ==1 (mod 3) and a number == {1,2} (mod 3) and therefore a number == {0,2} mod 3. This is never of the form 6n+4 ( == 1 mod 3). So the differences are in the class {0,2} mod 6, demonstrating that this is essentially a subsequence of A047238.]

Furthermore, differences 36, 144, 324,... of the form (6n)^2, A016910, appear in A047238 but not here, because prime(k)^2 -q=(6n)^2 is equivalent to prime(k)^2-(6n)^2 =q =(prime(k)+6n)*(prime(k)-6n), which requires an explicit factorization of the prime q. This is a contradiction if we assure that prime(k)-6n is not equal 1; if we scanned explicitly all primes up to prime(k)=10^7, for example, all (6n)^2 up to 6n<=10^7 are proved not to be in the sequence. (End)

Crossrefs

Cf. A000040 (primes), A001248 (primes squared).

Cf. A004277 (conjecturally, possible gaps between adjacent primes).

Cf. A054270 (prime below prime(n)^2).

Cf. A229489 (possible differences between prime(k)^2 and the next prime).

Programs

Mathematica

t = Table[p2 = Prime[k]^2; p2 - NextPrime[p2, -1], {k, 100000}]; Take[Union[t], 60]