A192224 P-integers: n such that the first phi(n) primes coprime to n form a reduced residue system modulo n, where phi is Euler's totient function A000010.

2, 4, 6, 12, 18, 30

Offset: 1

Author

Jonathan Sondow, Jun 29 2011

Keywords

• nonn
• fini
• full

Comments

Pomerance proved that the sequence is finite and conjectured that 30 is the largest element. Hajdu and Saradha proved Recamán's conjecture that 2 is the only prime P-integer. Both proofs use Jacobsthal's function A048669.

Hajdu, Saradha, and Tijdeman have a conditional proof of Pomerance's conjecture, assuming the Riemann Hypothesis.

Shichun Yanga and Alain Togbéb have proved Pomerance's conjecture. - Jonathan Sondow, Jun 14 2014

Examples

			12 is a P-integer because phi(12) = 4 and the first four primes coprime to 12 are 5, 7, 11, 13, which are pairwise incongruent modulo 12.
8 is not a P-integer because phi(8) = 4 and the first four primes coprime to 8 are 3, 5, 7, 11, but 3 == 11 (mod 8).
		

References

• B. M. Recamán, Problem 672, J. Recreational Math. 10 (1978), 283.

Links

• L. Hajdu, On a conjecture of Pomerance and the Jacobsthal function, 27th Journées Arithmétiques
• L. Hajdu and N. Saradha, On a problem of Recaman and its generalization
• L. Hajdu, N. Saradha, and R. Tijdeman, On a conjecture of Pomerance, arXiv:1107.5191 [math.NT], 2011.
• C. Pomerance, A note on the least prime in an arithmetic progression, J. Number Theory 12 (1980), 218-223.
• Shichun Yanga and Alain Togbéb, Proof of the P-integer conjecture of Pomerance, J. Number Theory, 140 (2014), 226-234. DOI: 10.1016/j.jnt.2014.01.014.

Crossrefs

Cf. A000010 (Euler totient function phi), A048669 (Jacobsthal function).