Saturday 9 March 2013

Sums of 2 square numbers

There are not many cases of 3 consecutive numbers where all three of them can be expressed as a sum of 2 squares
There are only 7 cases with numbers below 1000,which satisfy this requirement.These are shown below with the squares involved in each case shown within brackets-
72(36+36)..73(9+64)...74(25+49)
288(144+144)..289(64+225)..290(289+1)..This  of course can be rejected as one of the numbers 289 is itself a square.
520(196+324)...521(121+400)...522(81+441)
584(100+484)...585(144+441)..586(225+361)
800(400+400)...801(225+576)...802(361+441)
232(36+196)...233(64+169)...234(9+225)
808(324+484)...809(25+784)...810(81+729).
Taking into account cases similar to 288,289,290 which have to be rejected,there are 8 cases of numbers between 1000 and 2000 and  5 cases relating to numbers between 2000 and 3000.Can you find them?