The Pythagorean theorem carries the name of the Greek mathematician Pythagoras, who lived in the 6th century BCE, though the theorem had been known elsewhere for some time before. Although the Pythagorean theorem arose in geometry, we will be concerned strictly with the number theoretic properties of the Pythagorean equation, using the connection to geometry only as a jumping off point.

Theorem 1 (Pythagorean Theorem and converse)  Let x, y, and z be positive reals. Then z is the length of the hypotenuse of a right triangle with side lengths x, y, and z if and only if

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A Pythagorean triple is an ordered triple (x, y, z) of three positive integers such that x² + y² = z². If x, y, and z are relatively prime, then the triple is called primitive.

Let us first note the parity of x, y, and z in primitive triples, that is their values modulo 2. Since 0² ≡ 0, 1² ≡ 1, 2² ≡ 0, and 3² ≡ 1 mod 4, the only squares modulo 4 are 0 and 1. Letting X = x², Y = y², and Z = z², we have the following solutions to X + Y ≡ Z mod 4:

(1 + 1 ≡ 2 is not a solution because 2 is not a square modulo 4.) The first of these solutions corresponds only to nonprimitive triples — x, y, and z are all divisible by 2. Therefore any primitive triple corresponds to one of the other two solutions. In either case, Z is odd, and exactly one of X and Y is always odd. Thus z is odd, and exactly one of x and y is odd. To establish a convention, let us say that x is always odd and y is always even, since otherwise we can rename the variables of a given triple to obtain this. A primitive Pythagorean triple therefore has a unique representation (x, y, z), where y is even and x and z are odd.

Let a = z – x and b = z – y. Substituting for x and y into (1) gives

or In fact, only one of these values for z holds. The solution results in x or y being negative, so we take

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Define

Then the following hold:

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Solving for r and for s, we find

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The generators r and s will provide the basis of study for primitive triples. Most texts use the more common generators m and n, where x = m² – n², y = 2mn, and z = m² + n². Though I believe r and s as defined above are more natural choices (since they emphasize symmetry in the generators and not in the side lengths), the analysis is the same in the end, as m = r + s and n = r.

Theorem 2  A Pythagorean triple is primitive if and only if r and s are integers, s is odd, and (r, s) = 1.

Proof. Let (x, y, z) be a primitive Pythagorean triple (with even y). The quantities z + y and 2y are relatively prime because z and y are relatively prime in primitive triples and z + y is odd. We have

so z + y and z – y are relatively prime (because any common divisor would also divide 2y). The Pythagorean relation (1) implies that

so z + y and z – y must both be squares because they are relatively prime. Therefore is an odd integer. By (4), r is also an integer because

is even.
Now that we have established that r and s are integers for primitive triples, we will show that they are relatively prime. Let m = (r, s). Since m divides both r and s, m divides x, y, and z by (3). But since x and y (in particular) are relatively prime, this means m = 1.

For the converse, assume s is odd and (r, s) = 1. Because , we have that (r, z) = 1, since any divisor of both r and z is a divisor of s but r and s are relatively prime. By the definition of r, x + 2r² = z, so similarly (x, z) = 1. Therefore the triple (x, y, z) is primitive. ■

Corollary  The number of primitive Pythagorean triples is countably infinite.

Proof. Since each relatively prime pair (r, s) with odd s gives a unique primitive triple, it suffices to consider the cardinality of the set S = {(r, s) | r and s are relatively prime and s is odd}. This set is infinite since for each odd s there is at least one r relatively prime to s, but S is no larger than the rationals (by mapping S injectively into the rationals by sending (r, s) to r / s). Therefore the set S is countably infinite. ■

Next we will prove several results regarding triples modulo 3, 4, and 5.

Remark 1  If (x, y, z) is a primitive Pythagorean triple, then y ≡ 0 mod 4, z ≡ 1 mod 4, exactly one of {x, y} satisfies k ≡ 0 mod 3, and exactly one of {x, y, z} satisfies k ≡ 0 mod 5.

Proof. From (3), y = 2r (r + s). Either r is even or r + s is even (since s is odd), so 4 divides y.
As shown above, the only squares modulo 4 are 0 and 1. Since z = r² + (r + s)² is a sum of two squares and s is odd, there are two possibilities. If r is odd then z ≡ 1 + 0 ≡ 1 mod 4, and if r is even then z ≡ 0 + 1 ≡ 1 mod 4.

The only squares modulo 3 are 0 and 1. With X = x², Y = y², and Z = z², the congruence X + Y ≡ Z mod 3 has the following solutions:

The first case is nonprimitive, so either x² or y² is divisible by 3, and this implies that either x or y is divisible by 3.

The only squares modulo 5 are 0, 1, and 4, so X + Y ≡ Z mod 5 has solutions (up to permutation of X and Y):

Again, the first case is nonprimitive so exactly one side is divisible by 5. ■

To convince yourself that these are the only three generalizations we can make like this, experiment with different prime moduli with the Tripes Modulo p JavaScript program. Larger numbers (7, 11, 13,...) yield no obvious statements about the divisors of the respective triple members.

Definition  Two triples are siblings if they have a common hypotenuse.

The first such pair of primitive triples stem from the hypotenuse 65 (33² + 56² = 63² + 16² = 65²):

There are also primitive siblings of the hypotenuse 85:

If we search larger numbers, we can find larger sets of siblings. There are, for example, four primitive triples with hypotenuse 1105:

In general, the number of primitive Pythagorean triples of hypotenuse n is dependent on the number of prime factors of n that are congruent to 1 modulo 4. It turns out that only powers of 2 appear as these numbers. The numbers in the sequence 5, 65, 1105, 32045, 1185665, 48612265, 2576450045,... are the smallest hypotenuses that harbor 1, 2, 4, 8, 16, 32, 64,... primitive Pythagorean triples. The terms are given by multiplying successive primes that satisfy p ≡ 1 mod 4:

The hypotenuse 32045 has eight primitive triples:

But why do primitive triples form only in sets of 2^α, and what is the connection to primes p ≡ 1 mod 4? Let us look at the prime factors of n more closely.

By examining the data, it appears that the prime factors of each hypotenuse satisfy p ≡ 1 mod 4. Further, each prime hyptenuse holds only one triple, and the more factors a hypotenuse has, the more triples it holds. Specifically, if ω(n) is the number of distinct prime factors of n and all of the prime factors of n satisfy p ≡ 1 mod 4, then n appears to be the hypotenuse of exactly 2^{ω(n) – 1} primitive Pythagorean triples. (If n has any prime factors p ≡ 2 or 3 mod 4, then n is not the hypotenuse of any primitive triples.) In light of this, it makes sense that the first sibling sets of order 2^α will be found on the multiples of the first α primes that satisfy p ≡ 1 mod 4.

Of course, explaining where 2^{ω(z) – 1} arises is more tricky. Some experimentation shows that, for n with the above criteria, (3) has 2^{ω(n) – 1} positive integral solutions for r and s. We are interested in solutions to n = r² + (r + s)² where r and r + s are relatively prime (or, equivalently, (r, s) = 1). To see what is happening, first let us quote a theorem attributed to Fermat, the proof of which we leave to a book covering introductory number theory.

Theorem 3 (Fermat)  Let p be a prime such that p ≡ 1 mod 4. Then p can be expressed uniquely as a sum of two squares.

By "uniquely" it is meant that there exist unique integers a and b such that 0 < a < b and p = a² + b². Using Fermat's theorem, we decompose a number n into is prime factors and write each prime as a sum of two squares. We then make use of the Fibonacci identity,

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by which we can write the product of sums of two squares again as the sum of two squares. Multiplying together the sum of squares representations for the prime factors of n gives a sum of squares representation for n, and since the hypotenuse of a primitive triple is a sum r² + (r + s)² of two squares (by (3)), we will be able to generate primitive triples of hypotenuse n simply by knowing the sum of squares representations of the prime factors of n.

For example, for n = 1105,

This solution corresponds to r = 23, s = 1, which generate the triple (47, 1104, 1105). The other solutions to 1105 = r² + (r + s)² can be obtained by changing the order in which the factors are multiplied together and by reversing the sum of squares expressions of some factors.

Actually counting the solutions that we get in this way is cumbersome, however. The system is not a very natural one, and "Fibonacci multiplication" is in fact not even associative. Much better are the Gaussian integers — complex numbers of the form a + bi, where a and b are ordinary integers and i² = –1. By Fermat's theorem, each prime factor p ≡ 1 mod 4 of n conveniently has the representation p = a² + b² = (a + bi) (a – bi). Multiplying two Gaussian integers exactly results in the Fibonacci identity:

Rather than multiplying sum of squares representations, we will multiply Gaussian integers. We obtain the four solutions to 1105 = r² + (r + s)² by multiplying (Gaussian) "halves" of its (integer) factors p = (a + bi) (a – bi) . For each of the three primes dividing 1105, we have two choices of which Gaussian factor to take. Complex conjugation of a solution does not change the sum of squares representation, so we can choose the first such factor to have positive imaginary part. The solutions are then

By multiplying each by its complex conjugate, these correspond to the four representations of 1105 as a sum of two relatively prime squares (and thus to the four primitive triples with hypotenuse 1105).

Proposition  Let n > 1 be a natural number whose prime factors all satisfy p ≡ 1 mod 4. Then the number of distinct ways that n can be expressed as a sum a² + b² of two squares with a and b are relatively prime is 2^{ω(n) – 1}, where ω(n) is the number of distinct prime factors of n.

This result allows us to count triples of a given hypotenuse: Because z has 2^{ω(z) – 1} sum of squares representations when all its factors satisfy p ≡ 1 mod 4, there are 2^{ω(z) – 1} distinct solutions to (3). In each of these solutions, r and s are relatively prime (because r + si is not divisible by any integer prime) and s is odd (because an even s results in an even z). Therefore z is the hypotenuse of exactly 2^{ω(z) – 1} primitive Pythagorean triples. Not only does it allow counting, however, but it provides an algorithm to actually construct the triples given the sums of square representations of its prime factors. A simple program can readily compute, for example, the 2¹⁴ = 16384 primitive triples with hypotenuse n = 16880412096169215173498945, which is the product of the first fifteen primes that satisfy p ≡ 1 mod 4.

Until now we have considered triples based only on z; with what we have learned it is now easy to understand them sorted also by x or by y (where "P" denotes a prime leg and "P" denotes a prime power leg):

x		y	z	r	s		x		y	z	r	s		x	y		z	r	s		x	y		z	r	s
3	P	4	5	1	1		21		20	29	2	3		3	4	P	5	1	1		45	28		53	2	5
5	P	12	13	2	1		21		220	221	10	1		15	8	P	17	1	3		195	28		197	1	13
7	P	24	25	3	1		23	P	264	265	11	1		5	12		13	2	1		255	32	P	257	1	15
9	P	40	41	4	1		25	P	312	313	12	1		35	12		37	1	5		77	36		85	2	7
11	P	60	61	5	1		27	P	364	365	13	1		63	16	P	65	1	7		323	36		325	1	17
13	P	84	85	6	1		29	P	420	421	14	1		21	20		29	2	3		9	40		41	4	1
15		8	17	1	3		31	P	480	481	15	1		99	20		101	1	9		399	40		401	1	19
15		112	113	7	1		33		56	65	4	3		7	24		25	3	1		117	44		125	2	9
17	P	144	145	8	1		33		544	545	16	1		143	24		145	1	11		483	44		485	1	21
19	P	180	181	9	1

Very similar to the case of z, the prime factors of a given x determine how many triples have x as a leg (without the restriction that the prime factors must satisfy p ≡ 1 mod 4): If x ≡ 1 mod 2, then x is a leg of 2^{ω(x) – 1} primitive triples. And if y ≡ 0 mod 4, then y is a leg of 2^{ω(y) – 1} primitive triples.

Theorem 3  Let ω(n) be the number of distinct prime factors of a natural number n > 1. The number of primitive Pythagorean triples of which n is a member is

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We now generalize the problem of counting to include nonprimitive triples. (For a nonprimitive triple, r and s are irrational unless the common factor of the triple is a square.)

Theorem 4  If the natural number z decomposes into distinct primes as

with p_i ≡ 1 mod 4 for all i and q_j ≡ 3 mod 4 for all j, then z is the hypotenuse of exactly

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Pythagorean triples.

Proof. Let z be a natural number. Since primes q ≡ 3 mod 4 and the prime 2 do not contribute to primitive triples, these factors in z do not contribute to the total number of triples with hypotenuse z (i.e., removing all of these factors from a given z results in an integer that is the hypotenuse of the same number of triples as z). Therefore it suffices to consider natural numbers of the form with p_i ≡ 1 mod 4 for each i. The number of divisors of z that can be written as a product of powers of exactly j primes is σ_j(α₁,α₂,...,α_k), the jth elementary symmetric polynomial on the exponents α₁, α₂, ..., α_k. Each such divisor d of z is the hypotenuse of 2^{ω(d) – 1} = 2^{j – 1} primitive triples, by Theorem 3, and so contributes 2^{j – 1} nonprimitive triples of hypotenuse z by simply multiplying by z / d. Thus, by a property of the elementary symmetric polynomials,

■

For example, 3142875 = 3 · 5³ · 17² · 29 is the hypotenuse of exactly T(3142875) = ((2 · 3 + 1) (2 · 2 + 1) (2 · 1 + 1) – 1) / 2 = 52 Pythagorean triples.

Corollary  Let z = Q p₁ p₂ ... p_k > 1 be a natural number with exactly k distinct prime factors satisfying p ≡ 1 mod 4 and no factor of the form p² for any prime p ≡ 1 mod 4. Then the number of Pythagorean triples with hypotenuse z is T(z) = (3^k – 1) / 2.

In this section we will be interested in triples that satisfy certain conditions. Sometimes a triple that has a pair of sides with only a unit difference is referred to as a "twin." For example, the triples (3, 4, 5), (5, 12, 13), (7, 24, 25), and (21, 20, 29) each have a pair of sides that differ by 1. By browsing data, one finds that triples satisfying z – y = 1 are more common than those satisfying |x – y| = 1. There are infinitely many of both types, but the latter much less densely populate the world of primitive triples.

To find triples with b = z – y = 1, we solve x² + (z – 1)² = z² for z to find

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Any odd x then gives such a triple. Substituting this into the definition of r gives

Since s = √b = 1, the first few triples with b = 1 are:

Moreover, the sequence of y values is 4 times the triangular numbers, since, by (3),

This same procedure can be used for any valid a or b value because the resulting equation will always be linear in z. From (3) we see that the nth triples with fixed a and b values are given by

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We now turn to primitive triples with the restriction that |x – y| is a given difference d. The following is a list of the first few triples for which d = 1. (The first twenty-five of these triples are listed.)

To find the general form of these triples we must resort to a new technique, for the equation x² + (x + d)² = z² does not lead to a natural assignment of an index to one of the generators. But since r and s display simple patterns in fixed-a and -b triples, we might suspect that they will for fixed-d triples also. We notice two (among many) recurrence relations that seem to hold: r_n = r_{n – 1} + s_{n – 1} and s_n = r_n + r_{n – 1}. Substituting for s_{n – 1} we find

Second-order recurrence relations have solutions of the form r_n = kⁿ. Thus, from the recurrence relation,

the nonzero solutions of which satisfy k² – 2k – 1 = 0. So we have

Both of these values for k satisfy the recurrence, but we want the first and second terms to be 1 and 2 (the r values of the first two triples). Since a linear combination of the two values for k also satisfies the recurrence, we construct the general solution for r_n,

and solve for the constants we introduce by specifying the first two terms:

We find that

Then

and by the recurrence relations,

So the nth triple satisfying |x – y| = 1 is given (from (3)) by

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Clearly the difference of x_n and y_n is always 1. Intuitively, we know that as n gets large, the triples (x_n, y_n, z_n) approach the side lengths of a right isosceles triangle, the ratios z_n / x_n and z_n / y_n approaching √2. It is interesting, however, to examine the ratio s_n / r_n as n gets large; one finds that not only does this ratio also approach √2, but s_n / r_n is the nth continued fraction convergent to √2. This connection becomes apparent when we approach the solution of |x – y| = 1 triples another way and obtain a Pell equation, the solutions of which are these convergents:

or

Because the sequence s_n / r_n are the continued fraction convergents to √2, r_n and s_n are relatively prime. Moreover, s is odd, and so r and s generate a primitive triple.

When d ≠ 1, the situation is slightly more complicated. Take d = 7 for example. The first few triples are:

At first glance it appears that these r and s values hardly form nice recurrence relations. But in fact they do; there are two "threads" of the same relation (with different initial values) running through these triples, more easily seen when they are sorted:

x	y	z	r	s	x – y		x	y	z	r	s	x – y
5	12	13	2	1	–7		15	8	17	1	3	7
55	48	73	3	5	7		65	72	97	4	5	–7
297	304	425	8	11	–7		403	396	565	9	13	7
1755	1748	2477	19	27	7		2325	2332	3293	22	31	–7
10205	10212	14437	46	65	–7		13575	13568	19193	53	75	7

Again the recurrence is

Therefore, as above, the general solution is

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So the first thread of triples for which |x – y| = 7 is given by

and the second thread is given by

In general, x – y = s² – 2r², and the only numbers of this form, for r and s relatively prime and s odd, are those whose prime factors all satisfy p ≡ ±1 mod 8. These primes comprise the sequence 7, 17, 23, 31, 41, 47, 71, 73, 79, 89, 97,...  As representations as a sum of two squares were counted by factoring over Z[i] in Section III, we can count "fundamental" representations of a given number as s² – 2r² = (s + r√2) (s – r√2) by factoring over Z[√2]. For example, –1 = 1² – 2 · 1². The ring Z[√2] has infinitely many units, given by ±(1 + √2)ⁿ for integers n. Call a representation s² – 2r² fundamental if s ≤ r or s ≥ 2r. Each fundamental representation s₁² – 2r₁² gives rise to a thread of triples generated by s_n + r_n√2 = (s₁ + r₁√2) (1 + √2)ⁿ for nonnegative integers n. Each prime factor contributes two fundamental solutions, so we have the following result.

Proposition 2  Let d be a natural number whose prime factors all satisfy p ≡ ±1 mod 8. Then the number of fundamental representations of d in the form s² – 2r² for positive integers r and s is 2^ω(d), where ω(d) is the number of distinct prime factors of d.

(Alternatively, we could have considered sequences of triples that are infinitely long in both the positive and negative directions. In this case, there are 2^{ω(d) – 1} (double-sided) threads for a given d, and the case d = 1 must be separated from the others (since it still has only one thread, which is redundant). In this setup, however, we obtain triples with negative legs.)

Each fundamental solution (r₁, s₁) gives rise to the sequence of triples

For example, let d = 2737 = 7 · 17 · 23. The fundamental representations for these prime factors are

Thus 2737 has the following fundamental representations (where we divide each product by a power of 1 + √2 to obtain fundamental representations, in this case the first power):

An additional four threads come from the fundamental representations of –2737:

Relevant links:

• Pythagorean Triple on MathWorld
• Right Triangle on MathWorld
• NOVA: The Proof: NOVA's documentary about Andrew Wiles's proof of Fermat's Last Theorem
• Power Page: accurately subtitled "Interesting Stuff About Powers of Numbers"
• Pythagorean Triplets: all about triples that satisfy z – y = 1