Solving a Diophantine system
I am trying to find solutions to the following Diophantine system:
a+b+c=x^2
a^2+b^2+c^2=y^2
a^3+b^3+c^3=z^2
where a,b,c are less than 5000 and where x,y,z are perfect squares
1 Answer
Elementary version
sage: for a in srange(1, 5000):
....: for b in srange(1, a+1):
....: for c in srange(1, b+1):
....: if (a+b+c).is_square() and (a^2+b^2+c^2).is_square() and (a^3+b^3+c^3).is_square():
....: print(a,b,c)
A little bit more elaborate
sage: for abc in srange(1, 10000):
....: abc = abc*abc
....: for a in srange((abc + 2)// 3, abc):
....: for b in srange((abc - a + 1)//2, min(a, abc - a)):
....: c = abc - a - b
....: if (a^2+b^2+c^2).is_square() and (a^3+b^3+c^3).is_square():
....: print(a,b,c)
From which you get the list: (129, 124, 108), (516, 496, 432), (1161, 1116, 972), (2873, 2134, 34), (2064, 1984, 1728), (3225, 3100, 2700), ...
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Asked: 2020-09-12 23:00:29 +0100
Seen: 672 times
Last updated: Sep 13 '20
Should $x$, $y$, $z$ really be perfect squares?
So $a + b + c$, $a^2 + b^2 + c^2$, $a^3 + b^3 + c^3$ are perfect fourth powers?
Or did you mean $a + b + c$, $a^2 + b^2 + c^2$, $a^3 + b^3 + c^3$ are perfect squares?
If so, write one of the following:
Is this homework? Is this from Project Euler?
What have you tried? What specific problem(s) are you facing?
It is not for homework, I am doing research and I am just trying to find small numerical solutions without using elliptic curves.
I do mean that $a+b+c$, $a^2+b^2+c^2$, and $a^3+b^3+c^3$ are simply perfect squares, not perfect fourths.
And i have been trying to define each as a separate equation but I cant solve for the variables within a range as integers, something like:
for a in range(5000) for b in range(5000) for c in range(5000)
eq1 = a+b+c==x^2 eq2 = a^2+b^2+c^2==y^2 eq3 = a^3+b^3+c^3==z^2
solve([eq1,eq2,eq3],a,b,c,x,y,z)
I know this is not right, I am super new to Sage, please help!