A sequence based on consecutive primes

[Nick Hobson]

Hi Seqfans,

Suppose p and q are consecutive primes.  Let

   a = sqrt((p^2 + q^2)/2 - 1).   (1)

It is easy to see that if q = p + 2 then a is an integer.  Is the converse  
true?

If we consider the smallest prime q > p + 2 which makes a an integer, we  
generate the following sequence, where 0 is used if there is no such q:

11, 263, 59, 23, 101, 109, 0, 151, 193, 79, 269, 277, 311, 0, 179, 83,  
83003, 479, 487, 181, 563, 571, 613, 1201, 157, 141509, 739, 773, 479

So, for example, q = 11 is the smallest prime > 5 such that sqrt((3^2  
+ q^2)/2 - 1) is an integer.  There is no such q for p = 19 and p = 47  
(consider (1) modulo 7), but I can find no similar proof of impossibility  
for p = 127, which would generate the next term in the above sequence.

Source: M500 Magazine: http://www.m500.org.uk/

Nick