A187868 Primes of the form 100^k - 10^k + 1.

9901, 99990001, 999999000001, 9999999900000001

Offset: 1

Author

T. D. Noe, Mar 14 2011

Keywords

• nonn
• more

Comments

These primes correspond to k = 2, 4, 6, and 8. There are no other primes of this form for k up to 10000. These primes appear in A187614 because the decimal representation of their reciprocal contains only the digits 0, 1, 8, and 9.

Because these numbers are values of the sixth cyclotomic polynomial, by Theorem 1 of Golomb, the only k that can produce primes are the 3-smooth numbers, 2^i*3^j (A003586).

Next term has k > 1000000. - Robert Gelhar, Aug 20 2020

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 85.

Links

• Solomon W. Golomb, Cyclotomic polynomials and factorization theorems, Amer. Math. Monthly 85 (1978), 734-737.

Crossrefs

Cf. A003586, A168624, A187614.

Programs

Mathematica

Select[Table[100^n - 10^n + 1, {n, 1000}], PrimeQ]