A386996 Least positive integer m such that prime(2^m + 1) - prime(2^m) = 2*n.

1, 2, 4, 7, 24, 32, 44

Offset

1,2

Comments

Conjecture: a(n) exists for any positive integer n. In other words, we have {prime(2^m + 1) - prime(2^m): m = 1,2,3,...} = {2*n: n = 1,2,3,...}.

Note that de Polignac ever conjectured that for each positive integer n there are infinitely positive integers k such that prime(k + 1) - prime(k) = 2*n. Perhaps, for each n = 1,2,3,... there are infinitely many positive integers m such that prime(2^m + 1) - prime(2^m) = 2*n.

Daniel Weber at MathOverflow noted that a(8) > 78, a(9) = 11 and a(10) = 43. We also find that a(11) = 33, a(12) = 15, a(13) = 20 and a(15) = 13.

See a-file for all terms <= 78. - Michael S. Branicky, Oct 06 2025

Links

Table of n, a(n) for n=1..7.

Michael S. Branicky, Table of n, a(n), with values > 78 shown as -1

Zhi-Wei Sun, Is it true that {p_{2^m+1}-p_{2^m}: m in Z^+} = {2n: n in Z^+}?, Question 501247 at MathOverflow, Oct. 6, 2025.

Example

a(3) = 4 with prime(2^4 + 1) - prime(2^4) = 59 - 53 = 2*3.
a(4) = 7 with prime(2^7 + 1) - prime(2^7) = prime(129) - prime(128) = 727 - 719 = 2*4.
a(5) = 24 with prime(2^24 + 1) - prime(2^24) = 310248251 - 310248241 = 2*5.

Mathematica

p[n_]:=p[n]=Prime[n]; Do[m=1; Label[bb]; Do[If[p[2^m+1]-p[2^m]==2n, Print[n, " ", m]; Goto[aa]]; m=m+1; Goto[bb]]; Label[aa], {n, 1, 7}]

Crossrefs

Cf. A000040, A000079, A001359, A006512, A033844, A051439.

Sequence in context: A071795 A059501 A353986 * A138049 A099387 A359095

Adjacent sequences: A386993 A386994 A386995 * A386997 A386998 A386999

Keyword

nonn,more,hard

Author

Zhi-Wei Sun, Oct 06 2025

Status

approved