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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. 1

%I #31 Jan 22 2020 20:09:56

%S 2,4,6,12,18,30

%N 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.

%C 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.

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

%C Shichun Yanga and Alain Togbéb have proved Pomerance's conjecture. - _Jonathan Sondow_, Jun 14 2014

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

%H L. Hajdu, <a href="http://atlas-conferences.com/cgi-bin/abstract/cbbv-77">On a conjecture of Pomerance and the Jacobsthal function</a>, 27th Journées Arithmétiques

%H L. Hajdu and N. Saradha, <a href="http://www.math.klte.hu/~hajdul/recamanjntrev2.pdf">On a problem of Recaman and its generalization</a>

%H L. Hajdu, N. Saradha, and R. Tijdeman, <a href="http://arxiv.org/abs/1107.5191">On a conjecture of Pomerance</a>, arXiv:1107.5191 [math.NT], 2011.

%H C. Pomerance, <a href="http://dx.doi.org/10.1016/0022-314X(80)90056-6">A note on the least prime in an arithmetic progression</a>, J. Number Theory 12 (1980), 218-223.

%H Shichun Yanga and Alain Togbéb, <a href="https://doi.org/10.1016/j.jnt.2014.01.014">Proof of the P-integer conjecture of Pomerance</a>, J. Number Theory, 140 (2014), 226-234. DOI: 10.1016/j.jnt.2014.01.014.

%e 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.

%e 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).

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

%K nonn,fini,full

%O 1,1

%A _Jonathan Sondow_, Jun 29 2011

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