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Theorem 7.2: An integer $n$ can be represented as sum of two squares if and only if, it has factorization in the form:

$$n = 2^{\alpha} {p_1}^{\alpha_1} {p_2}^{\alpha_2} {p_3}^{\alpha_3} \ldots {p_r}^{\alpha_r} {q_1}^{\beta_1} {q_2}^{\beta_2} {q_3}^{\beta_3} \ldots {q_s}^{\beta_s},$$

where $p_i \equiv 1 \pmod 4$ and $q_j \equiv 3 \pmod 4$, $i = 1, 2, 3, \ldots, r$ and $j = 1, 2, 3, \ldots, s$, and all the $\beta_j$ exponents are even.

I'm having a big trouble to understand this theorem! Does the 2 indicate that the $n$ needs to be multiplied by it? I only know that by Theorem 7.1 a prime number $p$ in the form $4k + 1$ can be written as a sum of two squares.

Could anyone think of some examples where this theorem applies? I haven't found any example of it.

Robert Soupe
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Dalton
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  • $n=2^{\alpha}k$, are you talking about this $2$? $\alpha$ can be $0$, isn't? – MAN-MADE Aug 23 '17 at 06:41
  • Yes, this 2! I don't know about \alpha, that's the only information that I have. – Dalton Aug 23 '17 at 06:46
  • see my answer.... – MAN-MADE Aug 23 '17 at 06:48
  • The theorem applies to the numbers $1,2,4,5,8,9,10,13,16,17,18,20,25,26,29,32,34,36,37,40,\dots$ by asserting that each is a sum of two squares, and it applies to the numbers $3,6,7,11,12,14,15,19,21,22,23,24,27,28,30,31,33,35,38,39,\dots$ by asserting that each is not a sum of two squares. – Gerry Myerson Aug 23 '17 at 06:55
  • Speaking only for myself, I find the theorem interesting in its own right. It helps us recognize that numbers like 3, 7, 21, can't be represented as the sum of two squares, but a number like 45 can: $6^2 + 3^2 = 45$. This has applications for Waring's problem and Gaussian integers. – Robert Soupe Aug 25 '17 at 06:24

2 Answers2

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It means that, when $n$ is completely factored into primes, each prime divisor of the form $4k+3$ must occur an even number of times.

Example: $n=1170=2\cdot3\cdot3\cdot5\cdot13=2^1\cdot5^1\cdot13^1\cdot3^2=2^\alpha p_1^{\alpha_1} p_2^{\alpha_2}q_1^{\beta_1}$

where $p_1=5,\ p_2=13,\ q_1=3,\ \alpha=\alpha_1=\alpha_2=1,\ \beta_1=2.$

And sure enough, $1170=33^2+9^2=27^2+21^2.$

bof
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  • @skyking Oops! Thanks! – bof Aug 23 '17 at 07:05
  • I guess now I got it. So the only thing I need to check is: if there is a prime in a 4k+3 form that appears an even number of times then my n can't be written as sum of two squares, if doesn't then it can be written as sum of two squares? – Dalton Aug 23 '17 at 12:29
  • No. If there's a $4k+3$ prime that appears an odd number of times then $n$ isn't a sum of two squares. – Gerry Myerson Aug 23 '17 at 13:21
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    Read odd instead of even, sorry. – Dalton Aug 23 '17 at 15:03
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Any integer $n\neq 0$ can be written in the form $2^{\alpha}k$, where $k$ is a odd integer and $\alpha\geq 0$ i.e. $\alpha$ takes non-negative integer values.

Example:

$4=2^2$,

$5=2^0\cdot 5$,

$12=2^2\cdot 3$.

MAN-MADE
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