OFFSET
1,2
COMMENTS
Numbers can be parameterized as follows: choose an odd number a and k a divisor of (a^2+1)/2, k <= sqrt((a^2+1)/2) (otherwise you will get a duplicate value with a negative number being squared) and let d = a/k * (a^2+3ak+2k^2+1) / k. Then the three consecutive squares are (d-1)/2, (d+1)/2 and (d+3)/2. The squares added to them are (d/a + a) / 2, (d/a - a) / 2 and ((d+2)/(a+2k) - (a+2k))/2. All members of this sequence are congruent to 1 or 25 (mod 120); all are the product exclusively of primes congruent to 1 (mod 4). No number can written as the sum of two squares using four consecutive squares.
EXAMPLE
1105 = 31^2 + 12^2 = 32^2 + 9^2 = 33^2 + 4^2, using the consecutive values 31,32,33. 1 is included for the triple -1,0,1.
CROSSREFS
KEYWORD
nonn
AUTHOR
Franklin T. Adams-Watters, Oct 04 2006
STATUS
approved