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A little number theory fun. I am given that $167^2 + 32^2 = 28913$, and I am asked to find integers $a$ and $b$, such that $a^2 + b^2 = 28913000$.

Here's my thought process so far:

Knowing that $1000 = 10^2 + 30^2$, I rewrote $28913000$ as $28913\times 1000$, and proceeded to multiply the sums of squares:

$(167^2 + 32^2)(10^2 + 30^2).$

However, after foiling, I ended up with the sum of $4$ squares, and cannot think of a way to just find two squares, $a$ and $b$.

Any help would be greatly appreciated!

  • 2
    Hint: think of complex numbers, their multiplication, and their absolute-value-squared.2012-09-20

3 Answers 3

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Very good!

Do you know complex numbers? Assume that $-1$ has a square root somewhere (certainly not in $\mathbb R$), denote it $i$, and introduce $+$, $\cdot$ operations with reals and $i$. So, $i^2=-1$, thus $(a+bi)(a-bi) = a^2+b^2$. $(a+bi)(a-bi)(c+di)(c-di) = (a+bi)(c+di)\cdot (a-bi)(c-di)$ Can you calculate it?

  • 0
    Excellent! By treating the sum of squares in the reals as complex conjugates, the equation you wrote above provides a way to travel between the product of two sums of squares (the right side) to the sum of two squares (left side). Thanks!2012-09-20
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You can easily check that $ (a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2 $

  • 0
    See: [Brah$m$agupta–Fibo$n$acci identity](http://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity)2012-10-10
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One can also find all of the solutions computationally. E.g. here's some GAP code:

n:=28913000;;  for a in [0..RootInt(n)] do   b:=n-a^2;   c:=RootInt(b);   if(IsSquareInt(b)) then     Print(a^2+c^2," = ",a,"^2 + ",c,"^2\n");   fi; od; 

which gives

28913000 = 182^2 + 5374^2 28913000 = 710^2 + 5330^2 28913000 = 1330^2 + 5210^2 28913000 = 2062^2 + 4966^2 28913000 = 2174^2 + 4918^2 28913000 = 2630^2 + 4690^2 28913000 = 3370^2 + 4190^2 28913000 = 3766^2 + 3838^2 28913000 = 3838^2 + 3766^2 28913000 = 4190^2 + 3370^2 28913000 = 4690^2 + 2630^2 28913000 = 4918^2 + 2174^2 28913000 = 4966^2 + 2062^2 28913000 = 5210^2 + 1330^2 28913000 = 5330^2 + 710^2 28913000 = 5374^2 + 182^2