Find A B C so that A^2 + B^2 + C^2 = Square, A^3 + B^3 + C^3 = Cube. A Martin, 1898 Her answer: 11868013975030087 16269106368215226 88837226814909894 Smallest answer 14 23 70 A B C so that sums and differences are squares. 1873432, 2399057, 2288168 C Bumpkin 1750 Ladies Diary ------------------------------------------------------------------ >For a much harder problem, find integers A, B, and C so that >A^2 + B^2 + C^2 = X^2 and A^3 + B^3 + C^3 = Y^3. >An answer was published by A Martin in 1898. Keeping A, B and C under 500, there are 3 "native" cases: A B C X Y 3 34 114 119 115 14 23 70 75 71 18 349 426 551 493 All I did here was a ye olde loope. >Another toughie, solved by C Bumpkin in 1750: >Find A, B, C so that A+B, A-B, A+C, A-C, B+C and B-C >are all square numbers. Both of these were solved without >the aid of computers, but feel free to use one. Lowest 5 "natives": A B C squares of A+B ... B-C 434,657 420,968 150,568 ( 925, 117, 765, 533, 756, 520) 733,025 488,000 418,304 (1105, 495, 1073, 561, 952, 264) 993,250 949,986 856,350 (1394, 208, 1360, 370, 1344, 306) 2,399,057 2,288,168 1,873,432 (2165, 333, 2067, 725, 2040, 644) 2,843,458 2,040,642 1,761,858 (2210, 896, 2146, 1040, 1950, 528) 2,932,100 1,952,000 1,673,216 (2210, 990, 2146, 1122, 1904, 528)* * not native; interesting: has 3 squares equal to the one above it. This one was more fun. If A + B = X^2, then A - B = (X - U)^2 If A + C = Y^2, then A - C = (Y - V)^2 If B + C = Z^2, then B - C = (Z - W)^2 where U, V and W are any positive integers. Then, easy to show that: U, V and W are even, U > V and W > V Interesting equations, such as A = [2Y(Y-V) + V^2] / 2, can be developed, keeping "looping" reasonable. How in heck C.Bumpkin got a solution is simply amazing. Denis Borris Ottawa Ontario Canada ------------------------------------------------------------------ A,B,C.... Here are some solutions of sum of squares and cubes: ( GCD[a,b,c]=1) 1. 3^2 + 34^2 + 114^2 = 119^2 3^3 + 34^3 + 114^3 = 115^3 2. 14^2 + 23^2 + 70^2 = 75^2 14^3 + 23^3 + 70^3 = 71^3 3. 18^2 + 349^2 + 426^2 = 551^2 18^3 + 349^3 + 426^3 = 493^3 4. 145^2 + 198^2 + 714^2 = 755^2 145^3 + 198^3 + 714^3 = 721^3 This problem is much harder than "cubes&squares". Here is only one finding solution: A=434657 B=420968 C=150568 A-B=117^2 A-C=533^2 B-C=520^2 A+B=925^2 A+C=765^2 B+C=756^2 Best regards, NorT (Eugene V. Bryzgalov) ------------------------------------------------------------------------ Hi Ed Wow! Aren't they big numbers! Unfortunately, it is difficult to get enthusiastic about a problem if you know where to look up the answer! A Martin's A = 11868..., B = 16269..., C = 88837... C Bumpkin's A = 1873432, B = 2399057, C = 2288168. "C Bumpkin" was a pseudonym for some gentleman who did not want to show off - as in "Country Bumpkin", a simple farm yokel. When writing to the Ladies Diary or any other periodical, it was bad form to demonstrate how clever you were! If you look at Lewis Carroll's weekly puzzles, entrants were still using pseudonyms. I wonder how the others will get on. I realise now why the answer to your equation must be big. Best wishes John Gowland --------------------------------------------------------------------------