A339008 Least k such that p = k^2 + 1 and q = (k+2n)^2 + 1 are two consecutive prime numbers of the same form with q - p square.

24, 6, 312984, 16896, 240, 734994, 10640, 10360, 1946016, 2550, 13189264, 72996, 416520, 2184336, 1584360, 202484, 232696, 1700150, 2394456, 375360, 8736504, 9237866, 53629744, 360126, 87000, 574339974, 82404216, 23237760, 1249877496, 826650, 127119344, 1527720

Offset: 1

Author

Michel Lagneau, Nov 18 2020

Keywords

• nonn

Comments

4*n*(k + n) is a square. If n is a square, then k + n is also a square.

If n is prime, then n divides k.

a(n) = A339007(n) for n = 1, 2, 3, 4, 6, 7 and 9.

Examples

			a(1) = 24 because 24^2 + 1 = 577, (24 + 2)^2 + 1 = 677. The numbers 577 and 677 are two consecutive primes of the form m^2+1, and 677 - 577 = 10^2 is a square. The other values m such that p = m^2 + 1 and q = (m+2)^2 + 1 are consecutive primes with q - p square are 11024, 133224, 156024, 342224, 416024, ...
a(2) = 6 because 6^2 + 1 = 37, (6 + 4)^2 + 1 = 101. The numbers 37 and 101 are two consecutive primes of the form m^2+1, and 101 - 37 = 8^2 is a square. The other values m such that p = m^2 + 1 and q = (m+4)^2 + 1 are consecutive primes with q - p square are 16, 126, 1350, 1456, 1566, 2310, 5200, ...
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..60

Crossrefs

Cf. A002496, A096012, A193558, A206328, A216330, A339007.

Programs

Maple

for n from 1 to 25 do:
ii:=0:n1:=0:q:=2:
  for k from 2 by 2 to 10^9 while(ii=0) do:
    p:=k^2+1:
   if isprime(p)
    then
     x:=p-q:q:=p:z:=sqrt(x):
      if z=floor(z) and k-n1=2*n
       then
        ii:=1:printf(`%d %d \n`,n,n1):
         else
         n1:=k:
       fi:
    fi:
  od:
od:

PARI

consecutive(p, q) = {forprime(r = nextprime(p+1), precprime(q-1), if (isprime(r) && issquare(r-1), return(0));); return(1);}
a(n) = my(k=1); while (!(isprime(p=k^2+1) && isprime(q=(k+2*n)^2 + 1) && issquare(q-p) && consecutive(p, q)), k++); k; \\ Michel Marcus, Nov 30 2020

Extensions

a(26)-a(32) from Chai Wah Wu, Dec 06 2020