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
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>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
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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)
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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
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