Each of the n consecutive positive integers x+1, x+2, ...., x+n is expressible as the sum of squares of two distinct positive integers.
Determine the maximum value of n and prove that no higher value of n is possible.
(In reply to Not proven yet but I have a guess
.. due to Fermat:
A number N is expressible as a sum of 2 squares if and only if in the prime factorization of N, every prime of the form (4k+3) occurs an even number of times!
Examples: 245 = 5*7*7. The only prime of the form 4k+3 is 7, and it appears twice. So it should be possible to write 245 as a sum of 2 squares (in fact, try the squares of 14 and 7). But because 7 appears only once in 21=3*7, it is impossible to write 21 as the sum of two squares.
SO: Since 4K+3 in odd power CANNOT BE a sum of 2 squares , THEREFORE.....no two consecutive odd numbers are the sum of two distinct positive integers.- and 3 is the limit
REM; LAST SENTENCE CORRECTED after CHARLIE'S POST
Edited on November 9, 2012, 2:31 am