A306845 Sophie Germain primes which are Brazilian.

28792661, 78914411, 943280801, 7294932341, 30601685951, 919548423641, 2275869057821, 4172851565741, 4801096143881, 27947620155401, 29586967653101, 43573806645461, 119637719305001, 124484682222941, 148908227169101, 172723673300501

Offset: 1

Author

Bernard Schott, Mar 13 2019

Keywords

• nonn

Comments

These terms point out that the conjecture proposed in Quadrature "No Sophie Germain prime is Brazilian (prime)" (see link) was false.

Giovanni Resta has found the first counterexample of Sophie Germain prime which is Brazilian. It's the 141385th Sophie Germain prime 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73. The other counterexamples have been found by _Michel Marcus.

These numbers are relatively rare: only 25 terms < 10^15.

The 47278 initial terms of this sequence are of the form (11111)_b. The successive bases b are 73, 94, 175, 292, 418, 979, 1228, 1429, ...

The first term which is not of this form has 32 digits, it is 14781835607449391161742645225951 = 1 + 1309 + ... + 1309^9 + 1309^10 = (11111111111)_1309 with a string of eleven 1's. In this case, the successive bases b are 1309, 1348, 2215, 2323, 2461, ...

If (b^q - 1)/(b - 1) is a term, necessarily q (prime) == 5 (mod 6) and b == 1 (mod 3). The smallest term for each pair (q,b) is: (5,73), (11,1309), (17,1945), (23,20413), (29,5023), (41,9565), (47,2764) (See link Jon Grantham, Hester Graves).

Other smallest pairs (q, b) are: (53, 139492), (59, 154501), (71, 7039), (83, 9325), (89, 78028), (101, 8869), (107, 86503), (113, 89986), (131, 429226), (137, 929620), (149, 1954), (167, 175), (173, 1368025). - David A. Corneth, Mar 13 2019

Examples

			78914411 is a term because 2 * 78914411 + 1 = 157828823 is prime, so 78914411 is Sophie Germain prime, then, 78914411 = 1 + 94 + 94^2 + 94^3 + 94^4 = (11111)_94 and 78914411 is also a Brazilian prime.
		

Links

• Jon Grantham, Hester Graves, Brazilian Primes Which Are Also Sophie Germain Primes, arXiv:1903.04577 [math.NT], 2019.
• Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Crossrefs

Intersection of A005384 and A085104.

Cf. A007528, A306849.

Programs

PARI

lista(lim)=my(v=List(), t, k); for(n=2, sqrt(lim), t=1+n; k=1; while((t+=n^k++)<=lim, if(isprime(t) && isprime(2*t+1), listput(v, t)))); v = vecsort(Vec(v), , 8); \\ Michel Marcus, Mar 13 2019