# project-euler / project-euler / 75 / 75-analysis.txt

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39``` ```A^2 + B^2 = C^2 This already implies that C+A > B , C+B > A and A+B > C. The latter is because (A+B)^2 = A^2 + 2AB + B^2 > A^2+B^2 = C^2. Now, here is the modulo 2 of the results: A | B | === | C | A+B+C ----------------------- 0 0 0 | 0 1 0 1 | 0 1 1 0 | 0 ------------------------- Ergo, the sum is always even. (2a+1)*(2a+1) = 4a^2+4a+1 (2b+1)*(2b+1) = 4b^2+4b+1 [4a^2+4a+1+4b^2+4b+1] mod 4 = 2, and a square number cannot have a modulo of 2 when divided by 4. As a result, either one of 'a' or 'b' or both should be even. Now for mod 4: A | B | A^2 | B^2 | === | C^2 | C | A+B+C ------------------------------------------ 0 0 0 0 0 | 0 | 0 1/3 0 1 0 1 | 1 1 0 | 0 ------------------------- ------------------------------------------ If A^2 + B^2 = C^2 then so will (mA)^2 + (mB)^2 = (mC)^2 ------------------------------------------ A^2 + B^2 = C^2 ; A+B+C <= 1,500,000 ```