A083846 a(n) is the largest prime of the form x^2 + 1 <= 10^n.

5, 37, 677, 8837, 98597, 972197, 9985601, 99800101, 999444997, 9999200017, 99986234437, 999920001601, 9999799764517, 99999200001601, 999999202999697, 9999993200001157, 99999979750774757, 999999848000005777

Offset

1,1

Comments

It is conjectured that the number of primes of the form x^2+1 is infinite and thus this sequence does not become a constant, but this has not been proved. It is easily shown that all terms greater than 5 end in 1 or 7.

References

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

Links

Table of n, a(n) for n=1..18.

Eric Weisstein's World of Mathematics, Landau's Problems.

Mathematica

Do[ k = Floor[ Sqrt[ 10^n] - 1]; While[ !PrimeQ[k^2 + 1], k-- ]; Print[k^2 + 1], {n, 1, 19}]
lpf[n_]:=Module[{p=NextPrime[10^n, -1]}, While[!IntegerQ[Sqrt[p-1]], p= NextPrime[ p, -1]]; p]; Array[lpf, 10] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Feb 11 2023 *)

Crossrefs

Cf. A005574, A002496, A083844, A083845, A083847, A083848, A083849.

Sequence in context: A354862 A121834 A215233 * A358887 A336241 A180275

Adjacent sequences: A083843 A083844 A083845 * A083847 A083848 A083849

Keyword

nonn

Author

Harry J. Smith, May 05 2003

Extensions

Edited and extended by Robert G. Wilson v, May 08 2003

Status

approved