A235364 Twin primes p, p+2 such that p+1 is a Giuga number.

29, 31, 857, 859, 1721, 1723

Offset: 1

Author

Jonathan Sondow, Jan 07 2014

Keywords

• nonn
• more
• hard

Comments

For all 12 known Giuga numbers N, either both N-1 and N+1 are prime or neither is prime. Is it true that if any Giuga number N is adjacent to a prime N-1 or N+1, then in fact N lies between twin primes N-1, N+1?

See A235139 for a similar property of the known primary pseudoperfect numbers.

A007850 lists a 13th in its comments. - Bill McEachen, Jan 14 2014

If g = 420001794970774706203871150967065663240419575375163060922876441614\ 2557211582098432545190323474818 is confirmed as the 13th Giuga number, it will not be between a(7) and a(8), because g-1 is divisible by 13. So a(7) is not equal to g-1. But g+1 is prime (certified using the APRCL test in PARI) so g provides a negative answer to the above question. - Ralf Stephan, Jan 20 2014 (corrected by Jonathan Sondow, Jan 21 2014)

(Revision of my question.) For all 13 known Giuga numbers N, if N-1 is prime, then N+1 is also prime. Is it true that if any Giuga number N is 1 more than a prime, then N lies between twin primes N-1, N+1? - Jonathan Sondow, Mar 02 2014

Examples

			For the twin primes (p,p+2) = (29, 31), (857, 859), (1721, 1723), the numbers p+1 = 30, 858, 1722 are Giuga numbers (A007850).
		

Links

• MathWorld, Giuga Number
• Wikipedia, Giuga number

Crossrefs

Cf. A007850, A235139.