Fermat's theorem on sums of two squares

Summary

In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:

with x and y integers, if and only if

The prime numbers for which this is true are called Pythagorean primes. For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways:

On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4.

Since the Diophantus identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as the sum of two squares, by applying Fermat's theorem to the prime factorization of any positive integer n, we see that if all the prime factors of n congruent to 3 modulo 4 occur to an even exponent, then n is expressible as a sum of two squares. The converse also holds.[1] This generalization of Fermat's theorem is known as the sum of two squares theorem.

HistoryEdit

Albert Girard was the first to make the observation, describing all positive integer numbers (not necessarily primes) expressible as the sum of two squares of positive integers; this was published in 1625.[2][3] The statement that every prime p of the form 4n+1 is the sum of two squares is sometimes called Girard's theorem.[4] For his part, Fermat wrote an elaborate version of the statement (in which he also gave the number of possible expressions of the powers of p as a sum of two squares) in a letter to Marin Mersenne dated December 25, 1640: for this reason this version of the theorem is sometimes called Fermat's Christmas theorem.

Gaussian primesEdit

Fermat's theorem on sums of two squares is strongly related with the theory of Gaussian primes.

A Gaussian integer is a complex number   such that a and b are integers. The norm   of a Gaussian integer is an integer equal to the square of the absolute value of the Gaussian integer. The norm of a product of Gaussian integers is the product of their norm. This is Diophantus identity, which results immediately from the similar property of the absolute value.

Gaussian integers form a principal ideal domain. This implies that Gaussian primes can be defined similarly as primes numbers, that is as those Gaussian integers that are not the product of two non-units (here the units are 1, −1, i and i).

The multiplicative property of the norm implies that a prime number p is either a Gaussian prime or the norm of a Gaussian prime. Fermat's theorem asserts that the first case occurs when   and that the second case occurs when   and   The last case is not considered in Fermat's statement, but is trivial, as  

Related resultsEdit

Above point of view on Fermat's theorem is a special case of the theory of factorization of ideals in rings of quadratic integers. In summary, if   is the ring of algebraic integers in the quadratic field, then an odd prime number p, not dividing d, is either a prime element in   or the ideal norm of an ideal of   which is necessarily prime. Moreover, the law of quadratic reciprocity allows distinguishing the two cases in terms of congruences. If   is a principal ideal domain, then p is an ideal norm if and only

 

with a and b both integers.

In a letter to Blaise Pascal dated September 25, 1654 Fermat announced the following two results that are essentially the special cases   and   If p is an odd prime, then

 
 

Fermat wrote also:

If two primes which end in 3 or 7 and surpass by 3 a multiple of 4 are multiplied, then their product will be composed of a square and the quintuple of another square.

In other words, if p, q are of the form 20k + 3 or 20k + 7, then pq = x2 + 5y2. Euler later extended this to the conjecture that

 
 

Both Fermat's assertion and Euler's conjecture were established by Joseph-Louis Lagrange. This more complicate formulation relies on the fact that   is not a principal ideal domain, contrarily to   and  

AlgorithmEdit

There is a trivial algorithm for decomposing a prime of the form   into a sum of two squares: For all n such  , test whether the square root of   is an integer. If this the case, one has got the decomposition.

However the input size of the algorithm is   the number of digits of p (up to a constant factor that depends on the numeral base). The number of needed tests is of the order of   and thus exponential in the input size. So the computational complexity of this algorithm is exponential.

An algorithm with a polynomial complexity has been described by Stan Wagon in 1990, based on work by Serret and Hermite (1848), and Cornacchia (1908).[5]

DescriptionEdit

Given an odd prime   in the form  , first find   such that  . This can be done by finding a Quadratic non-residue modulo  , say  , and letting  .

Such an   will satisfy the condition since quadratic non-residues satisfy  .

Once   is determined, one can apply the Euclidean algorithm with   and  . Denote the first two remainders that are less than the square root of   as   and  . Then it will be the case that  .

ExampleEdit

Take  . A possible quadratic non-residue for 97 is 13, since  . so we let  . The Euclidean algorithm applied to 97 and 22 yields:

 
 
 
 
The first two remainders smaller than the square root of 97 are 9 and 4; and indeed we have  , as expected.

ProofsEdit

Fermat usually did not write down proofs of his claims, and he did not provide a proof of this statement. The first proof was found by Euler after much effort and is based on infinite descent. He announced it in two letters to Goldbach, on May 6, 1747 and on April 12, 1749; he published the detailed proof in two articles (between 1752 and 1755).[6][7] Lagrange gave a proof in 1775 that was based on his study of quadratic forms. This proof was simplified by Gauss in his Disquisitiones Arithmeticae (art. 182). Dedekind gave at least two proofs based on the arithmetic of the Gaussian integers. There is an elegant proof using Minkowski's theorem about convex sets. Simplifying an earlier short proof due to Heath-Brown (who was inspired by Liouville's idea), Zagier presented a non-constructive one-sentence proof in 1990.[8] And more recently Christopher gave a partition-theoretic proof.[9]

Euler's proof by infinite descentEdit

Euler succeeded in proving Fermat's theorem on sums of two squares in 1749, when he was forty-two years old. He communicated this in a letter to Goldbach dated 12 April 1749.[10] The proof relies on infinite descent, and is only briefly sketched in the letter. The full proof consists in five steps and is published in two papers. The first four steps are Propositions 1 to 4 of the first paper[11] and do not correspond exactly to the four steps below. The fifth step below is from the second paper.[12][13]

For the avoidance of ambiguity, zero will always be a valid possible constituent of "sums of two squares", so for example every square of an integer is trivially expressible as the sum of two squares by setting one of them to be zero.

1. The product of two numbers, each of which is a sum of two squares, is itself a sum of two squares.

This is a well-known property, based on the identity
 
due to Diophantus.

2. If a number which is a sum of two squares is divisible by a prime which is a sum of two squares, then the quotient is a sum of two squares. (This is Euler's first Proposition).

Indeed, suppose for example that   is divisible by   and that this latter is a prime. Then   divides
 
Since   is a prime, it divides one of the two factors. Suppose that it divides  . Since
 
(Diophantus's identity) it follows that   must divide  . So the equation can be divided by the square of  . Dividing the expression by   yields:
 
and thus expresses the quotient as a sum of two squares, as claimed.
On the other hand if   divides  , a similar argument holds by using the following variant of Diophantus's identity:
 

3. If a number which can be written as a sum of two squares is divisible by a number which is not a sum of two squares, then the quotient has a factor which is not a sum of two squares. (This is Euler's second Proposition).

Suppose   is a number not expressible as a sum of two squares, which divides  . Write the quotient, factored into its (possibly repeated) prime factors, as   so that  . If all factors   can be written as sums of two squares, then we can divide   successively by  ,  , etc., and applying step (2.) above we deduce that each successive, smaller, quotient is a sum of two squares. If we get all the way down to   then   itself would have to be equal to the sum of two squares, which is a contradiction. So at least one of the primes   is not the sum of two squares.

4. If   and   are relatively prime positive integers then every factor of   is a sum of two squares. (This is the step that uses step (3.) to produce an 'infinite descent' and was Euler's Proposition 4. The proof sketched below also includes the proof of his Proposition 3).

Let   be relatively prime positive integers: without loss of generality   is not itself prime, otherwise there is nothing to prove. Let   therefore be a proper factor of  , not necessarily prime: we wish to show that   is a sum of two squares. Again, we lose nothing by assuming   since the case   is obvious.
Let   be non-negative integers such that   are the closest multiples of   (in absolute value) to   respectively. Notice that the differences   and   are integers of absolute value strictly less than  : indeed, when   is even, gcd ; otherwise since gcd , we would also have gcd .
Multiplying out we obtain
 
uniquely defining a non-negative integer  . Since   divides both ends of this equation sequence it follows that   must also be divisible by  : say  . Let   be the gcd of   and   which by the co-primeness of   is relatively prime to  . Thus   divides  , so writing  ,   and  , we obtain the expression   for relatively prime   and  , and with  , since
 
Now finally, the descent step: if   is not the sum of two squares, then by step (3.) there must be a factor   say of   which is not the sum of two squares. But   and so repeating these steps (initially with   in place of  , and so on ad infinitum) we shall be able to find a strictly decreasing infinite sequence   of positive integers which are not themselves the sums of two squares but which divide into a sum of two relatively prime squares. Since such an infinite descent is impossible, we conclude that   must be expressible as a sum of two squares, as claimed.

5. Every prime of the form   is a sum of two squares. (This is the main result of Euler's second paper).

If  , then by Fermat's Little Theorem each of the numbers   is congruent to one modulo  . The differences   are therefore all divisible by  . Each of these differences can be factored as
 
Since   is prime, it must divide one of the two factors. If in any of the   cases it divides the first factor, then by the previous step we conclude that   is itself a sum of two squares (since   and   differ by  , they are relatively prime). So it is enough to show that   cannot always divide the second factor. If it divides all   differences  , then it would divide all   differences of successive terms, all   differences of the differences, and so forth. Since the  th differences of the sequence   are all equal to   (Finite difference), the  th differences would all be constant and equal to  , which is certainly not divisible by  . Therefore,   cannot divide all the second factors which proves that   is indeed the sum of two squares.

Lagrange's proof through quadratic formsEdit

Lagrange completed a proof in 1775[14] based on his general theory of integral quadratic forms. The following presentation incorporates a slight simplification of his argument, due to Gauss, which appears in article 182 of the Disquisitiones Arithmeticae.

An (integral binary) quadratic form is an expression of the form   with   integers. A number   is said to be represented by the form if there exist integers   such that  . Fermat's theorem on sums of two squares is then equivalent to the statement that a prime   is represented by the form   (i.e.,  ,  ) exactly when   is congruent to   modulo  .

The discriminant of the quadratic form is defined to be  . The discriminant of   is then equal to  .

Two forms   and   are equivalent if and only if there exist substitutions with integer coefficients

 
 

with   such that, when substituted into the first form, yield the second. Equivalent forms are readily seen to have the same discriminant, and hence also the same parity for the middle coefficient  , which coincides with the parity of the discriminant. Moreover, it is clear that equivalent forms will represent exactly the same integers, because these kind of substitutions can be reversed by substitutions of the same kind.

Lagrange proved that all positive definite forms of discriminant −4 are equivalent. Thus, to prove Fermat's theorem it is enough to find any positive definite form of discriminant −4 that represents  . For example, one can use a form

 

where the first coefficient a =   was chosen so that the form represents   by setting x = 1, and y = 0, the coefficient b = 2m is an arbitrary even number (as it must be, to get an even discriminant), and finally   is chosen so that the discriminant   is equal to −4, which guarantees that the form is indeed equivalent to  . Of course, the coefficient   must be an integer, so the problem is reduced to finding some integer m such that   divides  : or in other words, a 'square root of -1 modulo  ' .

We claim such a square root of   is given by  . Firstly it follows from Euclid's Fundamental Theorem of Arithmetic that  . Consequently,  : that is,   are their own inverses modulo   and this property is unique to them. It then follows from the validity of Euclidean division in the integers, and the fact that   is prime, that for every   the gcd of   and   may be expressed via the Euclidean algorithm yielding a unique and distinct inverse   of   modulo  . In particular therefore the product of all non-zero residues modulo   is  . Let  : from what has just been observed,  . But by definition, since each term in   may be paired with its negative in  ,  , which since   is odd shows that  , as required.

Dedekind's two proofs using Gaussian integersEdit

Richard Dedekind gave at least two proofs of Fermat's theorem on sums of two squares, both using the arithmetical properties of the Gaussian integers, which are numbers of the form a + bi, where a and b are integers, and i is the square root of −1. One appears in section 27 of his exposition of ideals published in 1877; the second appeared in Supplement XI to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie, and was published in 1894.

1. First proof. If   is an odd prime number, then we have   in the Gaussian integers. Consequently, writing a Gaussian integer ω = x + iy with x,y ∈ Z and applying the Frobenius automorphism in Z[i]/(p), one finds

 

since the automorphism fixes the elements of Z/(p). In the current case,   for some integer n, and so in the above expression for ωp, the exponent (p-1)/2 of -1 is even. Hence the right hand side equals ω, so in this case the Frobenius endomorphism of Z[i]/(p) is the identity.

Kummer had already established that if f ∈ {1,2} is the order of the Frobenius automorphism of Z[i]/(p), then the ideal   in Z[i] would be a product of 2/f distinct prime ideals. (In fact, Kummer had established a much more general result for any extension of Z obtained by adjoining a primitive m-th root of unity, where m was any positive integer; this is the case m = 4 of that result.) Therefore, the ideal (p) is the product of two different prime ideals in Z[i]. Since the Gaussian integers are a Euclidean domain for the norm function  , every ideal is principal and generated by a nonzero element of the ideal of minimal norm. Since the norm is multiplicative, the norm of a generator   of one of the ideal factors of (p) must be a strict divisor of  , so that we must have  , which gives Fermat's theorem.

2. Second proof. This proof builds on Lagrange's result that if   is a prime number, then there must be an integer m such that   is divisible by p (we can also see this by Euler's criterion); it also uses the fact that the Gaussian integers are a unique factorization domain (because they are a Euclidean domain). Since pZ does not divide either of the Gaussian integers   and   (as it does not divide their imaginary parts), but it does divide their product  , it follows that   cannot be a prime element in the Gaussian integers. We must therefore have a nontrivial factorization of p in the Gaussian integers, which in view of the norm can have only two factors (since the norm is multiplicative, and  , there can only be up to two factors of p), so it must be of the form   for some integers   and  . This immediately yields that  .

Proof by Minkowski's TheoremEdit

For   congruent to   mod   a prime,   is a quadratic residue mod   by Euler's criterion. Therefore, there exists an integer   such that   divides  . Let   be the standard basis elements for the vector space   and set   and  . Consider the lattice  . If   then  . Thus   divides   for any  .

The area of the fundamental parallelogram of the lattice is  . The area of the open disk,  , of radius   centered around the origin is  . Furthermore,   is convex and symmetrical about the origin. Therefore, by Minkowski's theorem there exists a nonzero vector   such that  . Both   and   so  . Hence   is the sum of the squares of the components of  .

Zagier's "one-sentence proof"Edit

Let   be prime, let   denote the natural numbers (with or without zero), and consider the finite set   of triples of numbers. Then   has two involutions: an obvious one   whose fixed points   correspond to representations of   as a sum of two squares, and a more complicated one,

 

which has exactly one fixed point  . Two involutions over the same finite set must have sets of fixed points with the same parity, and since the second involution has an odd number of fixed points, so does the first. Zero is even, so the first involution has a nonzero number of fixed points, any one of which gives a representation of   as a sum of two squares.

This proof, due to Zagier, is a simplification of an earlier proof by Heath-Brown, which in turn was inspired by a proof of Liouville. The technique of the proof is a combinatorial analogue of the topological principle that the Euler characteristics of a topological space with an involution and of its fixed-point set have the same parity and is reminiscent of the use of sign-reversing involutions in the proofs of combinatorial bijections.

This proof is equivalent to a geometric or "visual" proof using "windmill" figures, given by Alexander Spivak in 2006 and described in this MathOverflow post and this Mathologer YouTube video Why was this visual proof missed for 400 years? (Fermat's two square theorem) on YouTube.

Proof with partition theoryEdit

In 2016, A. David Christopher gave a partition-theoretic proof by considering partitions of the odd prime   having exactly two sizes  , each occurring exactly   times, and by showing that at least one such partition exists if   is congruent to 1 modulo 4.[15]

See alsoEdit

ReferencesEdit

  • D. A. Cox (1989). Primes of the Form x2 + ny2. Wiley-Interscience. ISBN 0-471-50654-0.*Richard Dedekind, The theory of algebraic integers.
  • L. E. Dickson. History of the Theory of Numbers Vol. 2. Chelsea Publishing Co., New York 1920
  • Harold M. Edwards, Fermat's Last Theorem. A genetic introduction to algebraic number theory. Graduate Texts in Mathematics no. 50, Springer-Verlag, NY, 1977.
  • C. F. Gauss, Disquisitiones Arithmeticae (English Edition). Transl. by Arthur A. Clarke. Springer-Verlag, 1986.
  • Goldman, Jay R. (1998), The Queen of Mathematics: A historically motivated guide to Number Theory, A K Peters, ISBN 1-56881-006-7
  • D. R. Heath-Brown, Fermat's two squares theorem. Invariant, 11 (1984) pp. 3–5.
  • John Stillwell, Introduction to Theory of Algebraic Integers by Richard Dedekind. Cambridge Mathematical Library, Cambridge University Press, 1996. ISBN 0-521-56518-9
  • Don Zagier, A one-sentence proof that every prime p ≡ 1 mod 4 is a sum of two squares. Amer. Math. Monthly 97 (1990), no. 2, 144, doi:10.2307/2323918

NotesEdit

  1. ^ For a proof of the converse see for instance 20.1, Theorems 367 and 368, in: G.H. Hardy and E.M. Wright. An introduction to the theory of numbers, Oxford 1938.
  2. ^ Simon Stevin. l'Arithmétique de Simon Stevin de Bruges, annotated by Albert Girard, Leyde 1625, p. 622.
  3. ^ L. E. Dickson, History of the Theory of Numbers, Vol. II, Ch. VI, p. 227. "A. Girard ... had already made a determination of the numbers expressible as a sum of two integral squares: every square, every prime 4n+1, a product formed of such numbers, and the double of the foregoing"
  4. ^ L. E. Dickson, History of the Theory of Numbers, Vol. II, Ch. VI, p. 228.
  5. ^ Wagon, Stan (1990), "Editor's Corner: The Euclidean Algorithm Strikes Again", American Mathematical Monthly, 97 (2): 125, doi:10.2307/2323912, MR 1041889.
  6. ^ De numerus qui sunt aggregata quorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 4 (1752/3), 1758, 3-40)
  7. ^ Demonstratio theorematis FERMATIANI omnem numerum primum formae 4n+1 esse summam duorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 5 (1754/5), 1760, 3-13)
  8. ^ 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, doi:10.2307/2323918, MR 1041893.
  9. ^ A. David Christopher. "A partition-theoretic proof of Fermat's Two Squares Theorem", Discrete Mathematics 339:4:1410–1411 (6 April 2016) doi:10.1016/j.disc.2015.12.002
  10. ^ Euler à Goldbach, lettre CXXV
  11. ^ De numerus qui sunt aggregata duorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 4 (1752/3), 1758, 3-40) [1]
  12. ^ Demonstratio theorematis FERMATIANI omnem numerum primum formae 4n+1 esse summam duorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 5 (1754/5), 1760, 3-13) [2]
  13. ^ The summary is based on Edwards book, pages 45-48.
  14. ^ Nouv. Mém. Acad. Berlin, année 1771, 125; ibid. année 1773, 275; ibid année 1775, 351.
  15. ^ A. David Christopher, A partition-theoretic proof of Fermat's Two Squares Theorem", Discrete Mathematics, 339 (2016) 1410–1411.

External linksEdit

  • Two more proofs at PlanetMath.org
  • "A one-sentence proof of the theorem". Archived from the original on 5 February 2012.{{cite web}}: CS1 maint: unfit URL (link)
  • Fermat's two squares theorem, D. R. Heath-Brown, 1984.