

A217503


Squared distance between consecutive primes of the form 4k+1 (see below).


2



1, 2, 2, 2, 2, 10, 8, 10, 8, 4, 2, 10, 4, 20, 18, 10, 2, 20, 58, 8, 40, 2, 40, 20, 10, 90, 2, 20, 10, 116, 2, 8, 20, 10, 2, 10, 20, 26, 4, 146, 8, 34, 10, 40, 34, 40, 2, 20, 2, 160, 50, 10, 180, 2, 180, 90, 58, 40, 130, 16, 116, 194, 50, 136, 74, 34, 52, 40
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Every prime p of the form 4k+1 has a unique solution p = x^2 + y^2. This sequence gives the squared distance between points (x,y) for consecutive primes of this form.
The squares mutual distance consecutive points with coordinates x(n) = A002331(n) and y(n) = A002330(n), where x(n)^2 + y(n)^2 = A002313(n) is prime.
Theorem: a(n) =/= A082073(n1) for all n > 1. Generally, it can be shown that there is no pair of primes p = a^2 + b^2 and q = x^2 + y^2 such that (a  x)^2 + (b  y)^2 = p  q > 0.


LINKS



EXAMPLE

5 = 1^2 + 2^2 and 13 = 2^2 + 3^2. The squared distance between the points (1,2) and (2,3) is 2, the second term of this sequence.


MATHEMATICA

nn = 200; p = Select[Prime[Range[nn]], Mod[#, 4] == 1 &]; q = {1, 1}; Table[pp = PowersRepresentations[p[[i]], 2, 2][[1]]; d = pp  q; q = pp; d[[1]]^2 + d[[2]]^2, {i, Length[p]  1}] (* T. D. Noe, Oct 19 2012 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



