A308816 - OEIS

%I #28 Feb 12 2021 12:14:32

%S 7,17,23,31,41,47,71,103,113,151,167,191,239,263,271,281,311,353,359,

%T 383,431,479,503,521,599,647,719,743,823,839,863,887,911,953,983,1031,

%U 1103,1151,1223,1319,1361,1367,1439,1487,1511,1559,1583,1607,1663,1823,1831,1847,1871,2039,2063,2087,2111

%N Primes r such that there exist primes p, q with p^2 - 2*q^2 = r.

%C If q is not 2 or 3, then r == 23 (mod 24).

%C The generalized Dickson conjecture implies that for any prime q there are infinitely many p for which p and p^2 - 2*q^2 are prime.

%C The least prime == 23 (mod 24) that is not in the sequence is 4079.

%C It appears that 5039 is also not in the sequence.

%H Jean-François Alcover, <a href="/A308816/b308816.txt">Table of n, a(n) for n = 1..586</a>

%H R. Israel et al., Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/3274493/p2-2-q2-5039-for-primes-p-q">p^2 - 2 q^2 = 5039 for primes p,q</a>

%H Robert Israel, <a href="/A308816/a308816_2.txt">Possibly incomplete list of 1043 members of the sequence</a>

%H Robert Israel, <a href="/A308816/a308816_1.txt">Numbers that may or may not be in the sequence</a>

%e a(3) = 23 is in the sequence because 11^2 - 2*7^2 = 23 and 11, 7 and 23 are all primes.

%p # Note: a "false" result from the filter procedure for prime r == 23

%p # (mod 24) is not definitive.  It is known that 4079 and 7703 are not in

%p # the sequence, but the status of 5039 has not been determined.

%p N:= 5000:

%p X1:= select(t -> isprime(t) and isprime(t^2-8), {seq(i,i=3..floor(sqrt(N+8)))}):

%p A1:= map(t -> t^2-8, X1):

%p X2:= select(t -> isprime(t) and isprime(t^2-18), {seq(i,i=3..floor(sqrt(N+18)))}):

%p A2:= map(t -> t^2-18, X2):

%p M:= <<3,2>|<4,3>>:

%p filter:= proc(r) local P1,S,C,k;

%p    if not isprime(r) then return false fi;

%p    P1:= select(t -> issqr(r+2*t^2), [$0..floor(2*sqrt(r))]);

%p    S:= map(t -> <sqrt(r+2*t^2),t>, P1);

%p    for k from 1 to 200 do

%p      C:= select(t -> isprime(t[1]) and isprime(t[2]), S);

%p      if C <> [] then return true fi;

%p      S:= map(t -> M . t, S);

%p    od;

%p    false

%p end proc:

%p A3:= select(filter, {seq(i,i=23..N,24)}):

%p sort(convert(A1 union A2 union A3, list));

%t (* This empirical program confirms that, for instance, 2791 (p=53,q=3), 3463 (p=59,q=3), 5023 (p=71,q=3) and 6871 (p=83,q=3), are in the sequence. *)

%t kmax[10463 | 10799 | 14543 | 21599 | 34871 | 45263] = 40;

%t kmax[r_] = 12; (* kmax(r) is the maximum of Solve parameter C[1] *)

%t s[r_, k_] := Solve[p > 1 && q > 1 && p^2 - 2 q^2 == r, {p, q}, Primes] /. C[1] -> k // FullSimplify // DeleteCases[#, {p -> Undefined, q -> Undefined}]&;

%t filterQ[r_] := Module[{k, srk}, For[k = 1, k <= kmax[r], k++, srk = s[r, k]; If[srk != {} && FreeQ[srk, ConditionalExpression], Print[{r, k, srk, p^2 - 2 q^2 /. srk}]; Return[True]]]; False];

%t Select[Prime[Range[2, 5000]], filterQ] (* _Jean-François Alcover_, Oct 19 2020 *)

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Jun 26 2019