A084165 Primes which are 1 mod m, where m is the index of the prime in sequence A002313 (Real primes with corresponding complex primes). The index m can be found in A084166 Primes which are -1 mod m can be found in sequence A084163.

5, 13, 17, 37, 89, 97, 181, 2689, 2969, 4621, 7457, 8081, 8161, 36709, 62701, 169489, 169709, 169753, 282809, 770101, 5763577, 9491101, 9491281, 9495121, 42544261, 115195501, 189689041, 189689653, 312315373, 312316409, 2294883817

Offset: 1

Author

Sven Simon, May 17 2003

Keywords

• nonn

Comments

Real primes 2,5,13,17,29,37,... have a unique representation as sum of two squares. Values larger 2 are the primes p with p = 1 mod 4. This is sequence A002313. If p = x^2 + y^2, the corresponding complex prime is x+y*i First complex prime is 1+i with 2 as corresponding real prime, according to reference, page 1-2.

Examples

			89 is the 11th prime in sequence A002313, 11*8 = 88, so 89 = 1 mod 11
		

References

• Handbook of First Complex Prime Numbers, Part1 + 2 Ervand Kogbetliantz and Alice Krikorian, Gordon and Breach, 1971

Crossrefs

Cf. A002313, A084163, A084166.

Programs

Mathematica

Module[{nn=112*10^6,pr,len},pr=Select[Prime[Range[nn]],MemberQ[ {1,2},Mod[ #,4]]&];len=Length[pr];Select[Thread[{pr,Range[len]}],Mod[ #[[1]],#[[2]]] == 1&]][[All,1]] (* Harvey P. Dale, Aug 13 2020 *)