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A New Approach to the Girard-Fermat’s Two Squares Theorem with Some Remarks on Primes’ Decomposition as Sums of Squares

Arto Adili*, Adrian Naço, Lorena Kelo

Department of Mathematics and Physics, “Fan S. Noli” University, Korçë, Albania

Abstract: A new elementary approach to the Girard-Fermat theorem, generalizing the equality of the sum of squares with the prime to its divisibility by that prime. That reveals some interesting properties of the corresponding decomposition’s components.

Keywords: Girard-Fermat theorem, primitive root, indexes

Received February 14, 2026; Revised March 16, 2026; Accepted March 23, 2026

Citation: Arto Adili, Adrian Naço, Lorena Kelo, A New Approach to the Girard-Fermat’s Two Squares Theorem with Some Remarks on Primes’ Decomposition as Sums of Squares. Turkish Journal of Analysis and Number Theory, 14 (1). 1-5. January 2026. doi:10.12691/tjant-14-1-1

1. Introduction

The issues of the numbers which are sum of two squares has been addressed early. It has been studied the amount of different representations of the same number as a sum of squares, and the conditions that a number has such representations. Some first results, albeit accompanied by uncertainly and inaccuracy, as quoted by Dixon [1] pg. 225, are mentioned by Diofant and Mohammed Ben Alhocain (Arab mathematician in tenth century). The first exact formulations of the existence of such representations are mentioned at [1] pg. 227.

The earliest one is that of Albert Girard (1625) - “a number is a sum of two squares only when: It is a square, it is a prime of the form , it is the product of such numbers, and when it is the double of one of the foregoing”. Fermat (1640) wrote that the condition for a number of the form to be a sum of two squares, the number must not be odd, and , when divided by the greatest square entering it as a factor, must not be divisible by any prime of the form (as quoted by [1]). Fermat (quoted by Dickson [1], pg. 228) calls the proposition staying in the core of their formulations: “Every prime number is a sum of two squares” the “fundamental theorem on right triangles”. Dickson himself calls it “Girard theorem”.

Generally, the proposition:

Theorem 1.1. An odd prime can be expressed as

with and integers, if and only if .

is called “Fermat’s theorem on sums of two squares”. The prime numbers for which this is true are called Pythagorean primes.

Its proof has a long history too. Fermat asserts (as Dickson [1] quoted at pg. 228) that it has an irrefutable proof, based on the “method of indefinite descent”. The first known irrefutable proof is that of Euler (1747, 1749), another one is that of Lagrange (1775), simplified by Gauss. Dedekind has provided two ones, which use the ring of Gaussian integers; are also well known two quite different and almost recent ones by Heath Brown and Zagier [2].

We’ll call it “the Girard-Fermat’s two-square theorem”.

The uniqueness of prime’s representation as sum of squares was observed early by Euler, which, for the numbers that appeared in two different ways as sums of squares, offered a factorization method, leading to an factorizing algorithm running in steps [3].

The following results will be used in our approach to the demonstration of the Girard-Fermat theorem.

Theorem 1.2. [4] If is a prime number such that , then or .

Theorem 1.3. [5] Let be a polynomial with integer coefficients and the canonical factorization of the positive integer . The number of roots of the equation

equals the product of the roots’ numbers of the following equations

2. Some properties of couple such are multiples of a prime

Let be a prime of the form . In this section we find all the pairs of positive integers , , such that and some their interesting properties.

We will use the following proposition:

Theorem 2.1. [6] If is a prime , then for every divisor of the congruence has distinct solutions smaller than .

If the prime is , then for some . So, for the divisor of theorem 2.1. yields that the congruence , has distinct solution. Let as denote them by . (∗)

Theorem 2.2. [7] Consider the congruence

for some . If is a primitive root modulo and , then the given congruence is equivalent to the congruence

This congruence is linear regarding and admits solutions if is divisible by . If this is the case, it has distinct solutions.

The prime number has primitive roots, let be one of them. For each at (∗) , let us consider the congruence

These congruences satisfy the conditions of Theorem 2.2., for , , so they are equivalent to the corresponding

Necessary and sufficient condition for these linear congruences to admit any solution is that

.

is a primitive root modulo , so . Also, by we get

Taking into account that , , we have

which means that

being a primitive root modulo . This way we have that

.

So, even the indexes must be multiples of 4. Since , the congruences

have solutions, then each one of them has 4 distinct ones.

Let us consider the first two positive solutions of each one of the following equations:

All of them are in the set . Taking into account that there are such congruences, each one of them having two distinct solutions in the set , then the numbers

can be organized in couples such that for each one of them

, .

For the positive integers and we have:

wherefrom

because , with and such .

We have proved the following:

Theorem 2.3. For every prime , the set

can be divided in couples of numbers, the sum of squares of each of them being divisible by the prime .

Example 2.4. For the prime we have the following “decompositions” in sums of two squares:

So, the numbers are divided in 7 couples, the sum of squares of each one of them is divisible by 29.

In the following, we will can two couples of natural numbers and congruent modulo , , if and write

We put also the following “scalar multiplication” in the set of couples

Definition 2.5. Let be a prime number. We will call as “a square couple of the prime ” every couple of non-negative integers such that . A square couple of is called a “primitive square couple” if .

In the following, we will not make any distinction between the couples and , considering them the same couple.

Theorem 2.6. For every prime there are exactly primitive square couples of , generated by multiplying any arbitrary primitive square couple by elements of a reduced residue system modulo .

Proof. Since for every the congruence

has exactly two solutions in this set, then for every such exist exactly two primitive square couples of which have the number . But the number of elements of the set is and every number appears twice in these primitive square couples, so there are exactly primitive square couples of the prime

Let be any primitive square couple of the prime and the respective reduced residue system. Let us now consider the following primitive square couples:

They are all different from each other. Indeed, if

then

Example 2.7. is a primitive square couple of the prime . Using the construction of the proof of theorem 2.6. we get the following list of all primitive square couples of 13:

Corollary 2.8. For every two square couples and of the prime , there is a positive integer such that:

Theorem 2.9. If and are squares couples of the prime , the for every ,

is also a square couple of the prime .

Proof. Corollary 2.8. yields a positive integer , such that , namely

and .

Since

we have . So, the couple

is a square couple of prime .

Remark 1. We have these special cases:

● For . If and are square couples of prime , then is also a square couple of the prime .

● For . If and are square couples of prime , then is also a square couple of the prime .

Theorem 2.10. Let be a prime . Then the sums of the components of the primitive square couples of form a reduced residue system modulo .

Proof. Firstly, we show that if is a primitive square couple of , then . Indeed, if

then , consequently , which is impossible because .

Since there are primitive square couples, to prove the theorem it suffices to show that for every two primitive square couples the sums of their components are not congruent modulo . Suppose that and are two primitive square couples such that

By Corollary 2.8. exists such that and . So, we have

which, taking into account that , means that and , are not different.

3. Proof(s) of Girard-Fermat Theorem

In this section we will present two new proofs of Girard-Fermat Theorem, relying on the results of the section 2.

Theorem 3.1. (Girard-Fermat) Every prime number of the form can be expressed as a sum of two squares of two natural numbers in only one way.

Proof. Let a primitive square couple of the prime with the least sum of its components, whose existence is provided by Theorem 2.10. We will now prove that .

First proof. Let as suppose that . Since is a primitive square couple, then for any natural and by Theorem 1.2. all the divisors of are .

By Theorem 1.3. the number can be expressed as sum of two squares in at least two ways:

Supposing that and we have:

By the supposition , so

and, taking into account that we obtain that

We denote by . We have to prove that . If then the equality

gives

and ,

whereby the contradiction:

If , than the couples of numbers , and , have the same parity, so the halves , , , , are integers. Similarly, we get that the equalities

and

yield

and ,

whereby the contradiction:

.

So, and is a primitive square couple whose sum of components is:

,

which contradicts our supposition .

Second proof. Firstly, we see that . Indeed, if , then is also a primitive square couple, because

and .

But the sum of elements of this couple is , which is impossible. The positive integer and can’t be both odd. Indeed, if we get , since:

and

the couple is a primitive squares couple. So, the sum of elements of this couple is

and , which is impossible. Also, the positive integers and cannot be both even, because . Consequently, a primitive square couple of prime with the smallest sum of its elements has one component an odd number and the other an even number.

Suppose now that . Since , there is any integer such that . But, from Theorem 1.2. the divisors of are all . By Theorem 1.3., the number can be expressed as sum of two squares in at least two ways:

We see that the parities of and must be different, because those of and are different. Suppose that and . Then .

Since we have , then

Now, since and are square couples of the prime , by Theorem 1.2. we have that:

is also a square couple of the prime . We noticed that the parities of and are different and the same holds for and , so the components of the couple have the same parity. The following two cases are possible:

Case 1. The numbers and are both odd. Since , there is a square couple

with sum of its elements . But, since:

then , wherefrom

which contradicts the assumption.

Case 2. The numbers and are both even. The couple

is a square couple and the sum of its elements is

But, since we have:

which also contradicts the assumption.

References

[1]   | Dickson, Leonard Eugene (2005) [1920] - History of the theory of numbers. Vol. II: Diophantine analysis, New York: Dover Publications, ISBN 978-0-486-44233-4.
[2]   | Zagier, D. (1990) A one-sentence proof that every prime p ≡ 1 mod 4 is a sum of two squares, American Mathematical Monthly, 97 (2): 144.
[3]   | James McKee Turning Euler’s factoring method into a factoring algorithm Bull. Lon don Math. Soc. 28 (1996) 351-355.
[4]   | Sato N.: Art of Problem Solving (Notes on Number Theory for students at the IMO), https:// artofproblemsolving.com/ articles/files/SatoNT.pdf (accessed at February 17, 2019).
[5]   | Sierpinski W.: Elementary theory of Numbers, North-Holland, PWN-Polish Scientific Publisher.
[6]   | Adler A., Coury J.E.: The Theory of Numbers, Jones and Barlett Publisher, Boston, London, Singapore, 1995.
[7]   | Apostol T.M.: Introduction to Analytic Number Theory, Springer-Verlag 1976.