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Prime numbers that occur as divisors of numbers of the form m^2 + 5.
3

%I #35 Nov 29 2016 03:27:01

%S 2,3,5,7,23,29,41,43,47,61,67,83,89,101,103,107,109,127,149,163,167,

%T 181,223,227,229,241,263,269,281,283,307,347,349,367,383,389,401,409,

%U 421,443,449,461,463,467,487,503,509,521,523,541,547,563

%N Prime numbers that occur as divisors of numbers of the form m^2 + 5.

%C Conjecture: a prime number is in this sequence if and only if its next-to-last digit is even.

%C The law of quadratic reciprocity shows an odd prime is in the sequence if and only if it is 1, 3, 5, 7 or 9 (mod 20). This proves the above conjecture, so the sequence is the union of {2, 5} and A139513. - _Jens Kruse Andersen_, Aug 09 2014

%H Jens Kruse Andersen, <a href="/A240920/b240920.txt">Table of n, a(n) for n = 1..10000</a>

%H David Lowry-Duda, <a href="http://www.jstor.org/stable/10.4169/college.math.j.46.1.56">Unexpected Conjectures about -5 Modulo Primes</a>, College Mathematics Journal, Vol. 46, No. 1 (Jan 2015), pp.56-57.

%F a(n) ~ 2n log n. - _Charles R Greathouse IV_, Nov 29 2016

%e 23 is in the sequence because it divides 8^2+5=69 with m=8.

%p isA240920 := proc(p)

%p local n;

%p if isprime(p) then

%p for n from 0 to p do

%p if modp(n^2+5,p) = 0 then

%p return true;

%p end if;

%p end do:

%p false;

%p else

%p false;

%p end if;

%p end proc:

%p for i from 1 to 600 do

%p p := ithprime(i) ;

%p if isA240920(p) then

%p printf("%d,",p);

%p end if;

%p end do:

%o (PARI) select(p->issquare(Mod(-5,p)), primes(100)) \\ _Charles R Greathouse IV_, Nov 29 2016

%Y Cf. A002313 (k=1 or k=4), A033203 (k=2), A045331 (k=3), A139513.

%K nonn,easy

%O 1,1

%A _J. Lowell_, Aug 02 2014