The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A297620 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A297620

Positive numbers n such that n^2 == p (mod q) and n^2 == q (mod p) for some consecutive primes p,q.
(history; published version)

#12 by Peter Luschny at Tue Jan 02 04:38:03 EST 2018

STATUS

proposed

approved

#11 by Robert Israel at Tue Jan 02 02:00:43 EST 2018

STATUS

editing

proposed

#10 by Robert Israel at Tue Jan 02 02:00:35 EST 2018

LINKS

Robert Israel, <a href="/A297620/b297620.txt">Table of n, a(n) for n = 1..2305</a>

#9 by Thomas Ordowski at Tue Jan 02 01:58:06 EST 2018

EXAMPLE

a(3) = 12 is in the sequence because 71 and 71 73 are consecutive primes with 12^2 == 73 (mod 71) and 12^2 == 71 (mod 73).

STATUS

proposed

editing

#8 by Altug Alkan at Mon Jan 01 22:54:29 EST 2018

STATUS

editing

proposed

#7 by Altug Alkan at Mon Jan 01 22:53:29 EST 2018

COMMENTS

Odd terms of this sequence are 93, 145, 187, 215, 241, 297, 317, 329, 349, 357, 423, 431, 535, ... - Altug Alkan, Jan 01 2018

#6 by Altug Alkan at Mon Jan 01 22:52:01 EST 2018

COMMENTS

Odd terms of this sequence are 93, 145, 187, 215, 241, 297, 317, 329, 349, 357, 423, 431, ... - Altug Alkan, Jan 01 2018

STATUS

proposed

editing

#5 by Robert Israel at Mon Jan 01 21:59:57 EST 2018

STATUS

editing

proposed

#4 by Robert Israel at Mon Jan 01 20:51:26 EST 2018

MAPLE

newR:={seq(seq(chrem([sp*xp, sq*xq], [p, q]), sp=[-1, 1]), sq=[-1, 1])};

#3 by Robert Israel at Mon Jan 01 20:47:50 EST 2018

COMMENTS

Positive numbers n such that n^2 == p+q mod (p*q) for some consecutive primes p, q.

Each pair of consecutive primes p,q such that p is a quadratic residue mod q and p and q are not both == 3 (mod 4) contributes infinitely many members to the sequence.

MAPLE

N:= 1000: # to get all terms <= N

R:= {}:

q:= 3:

while q < N^2 do

p:= q;

q:= nextprime(q);

if ((p mod 4 <> 3) or (q mod 4 <> 3)) and numtheory:-quadres(q, p) = 1 then

xp:= numtheory:-msqrt(q, p); xq:= numtheory:-msqrt(p, q);

newR:={seq(seq(chrem([sp*xp, sq*xq], [p, q]), sp=[-1, 1]), sq=[-1, 1])};

for sp in [-1, 1] do for sq in [-1, 1] do

v:= chrem([sp*xp, sq*xq], [p, q]);

R:= R union {seq(v+k*p*q, k = 0..(N-v)/(p*q))}

od od;

fi;

od:

sort(convert(R, list));