1. On 25 Dec. 1640, P. de Fermat (1607-1665) wrote to M. Mersenne, OM, about the two squares theorem for prime numbers, called Fermat's Christmas theorem.
Only some numbers can be written as the sum of two squares. P. de Fermat began a line of enquiry, founded on / premised upon the prime numbers, into which ones can be so written, on 25.12.1640; this can be considered the beginning of the modern higher arithmetic.
He stated in tbe letter that among (odd) primes, those (and, obviously, as shown below (art. 2), only those) of the form p = 1 (mod 4) can be so written (i. e., those primes which yield 1 as remainder/residue on dividing by 4).
e. g.:
5 = 1 (mod 4).
5 = 1^2 + 2^2.
Call such (odd) primes Pythagorean primes, since the sum of two squares is reminiscent of the Pythagoras theorem; in contradistinction, call (odd) primes of the form p = 3 (mod 4) = -1 (mod 4) Gaussian primes (in the integers Z) (the reason for the name is explained in art. 6 below).
Unlike 5 (a Pythagorean prime); 3, 7, or 11, all Gaussian primes, cannot be so written; 13 = 2^2 + 3^2 = 1 (mod 4), a Pythagorean prime.
N. B.: If a prime is written as the sum of two numbers, they, the addends, are relatively prime.
2. For (odd) primes which can be so written, it is clear that they must be written as the sum of an odd square and an even square.
Odd squares are of the form x = 1 (mod 4); Even squares, of the form y = 0 (mod 4). Thus their sum is of the form z = 1 (mod 4).
Thus all such sums are of the form z = 1 (mod 4), and only such primes, the Pythagorean primes, can be so written. The Gaussian primes cannot be so written.
What is not obvious is that all the Pythagorean primes can be so written. This is what P. de Fermat discovered and stated on 25.12.1640.
3. P. de Fermat did not give a demonstration of the propositon he had enunciated.
A complete demonstration of his Christmas theorem was given by L. Euler (1707-1783) only more than a century later, in two letters to C. Goldbach on 6.5.1747 and 12.4.1749, later published in two articles `De numeris qui sunt aggregata duorum quadratorum' [E228] and `Demonstratio theorematis Fermatiani omnem numerum primum formae 4n+1 esse summam duorum quadratorum' [E241].
J. L. Lagrange {G. L. La Grangia} (1736-1813) gave another demonstration in `Recherches d'arithmetique', Berlin, 1773, from his study of binary quadratic forms; C. F. Gauss (1777-1855) gave a succinct one in art. 182 of his `Disquisitiones arithmeticae', 1801.
4. A. Girard (1595-1632) had stated, in 1625, with no demonstrations, that the numbers that can be written as the sum of two squares are:
i. 2 = 1^2 + 1^2 (the even prime).
ii. Squares. (N. B.: 0^2 is also permissible in the sum.)
iii. Primes of the form p = 1 (mod 4), the Pythagorean primes.
iv. Products of such numbers.
(in his commentary and annotated edition of the Arithmetic of Simon Stevinus).
5. Diophantus of Alexandria (fl. IIIrd century) showed (in his `Arithmetica') that if two numbers can be written as the sum of two squares, then so can be their product (and, in two different ways).
(a^2 + b^2) (c^2 + d^2) = |a + bi|^2 |c + di|^2
= (a+bi) (a-bi) (c+di) (c-di)
= (a+bi) (c+di) (a-bi) (c-di)
= [(ac-bd) + (ad+bc)i] [(ac-bd) - (ad+bc)i]
= |(ac-bd) + (ad+bc)i|^2
= (ac-bd)^2 + (ad+bc)^2
(a^2 + b^2) (c^2 + d^2) = (a^2 + b^2) (d^2 + c^2)
= |a + bi|^2 |d + ci|^2
= (a+bi) (a-bi) (d+ci) (d-ci)
= (a+bi) (d+ci) (a-bi) (d-ci)
= [(ad-bc) + (ac+bd)i] [(ad-bc) - (ac+bd)i]
= |(ad-bc) + (ac+bd)i|^2
= (ad-bc)^2 + (ac+bd)^2
= (ac+bd)^2 + (ad-bc)^2
Thus we obtain that:
(a^2 + b^2) (c^2 + d^2) = e^2 + f^2
where:
either: e = ac-bd; f = ad+bc
or: e = ac+bd; f = ad-bc.
N. B.: Diophantus merely stated his identity, and gave no demonstration. I have written an anachronistic demonstration for his identity in the ring of Gaussian integers Z[i] (cf. the following art. 6 and, further, art. 9).
6. C. F. Gauss in his second memoir on biquadratic {quartic} reciprocity,
`Theoria residuorum biquadraticorum', 1831, discovered the ring Z[i] of
Gaussian integers of the form a+bi (where a and b are integers, and the lateral/imaginary unit i =
\sqrt{-1} is adjoined to the integers Z to obtain Z[i]; cf. art. 10 on the lateral unit), and showed that
unique factorisation (the fundamental theorem of arithmetic) holds. The
Pythagorean primes in Z, of the form p = 1 (mod 4), split into conjugate
primes in the ring Z[i] (reflecting their being written as the sum of
two squares) (cf. art. 11 for a demonstration that the factors are Gaussian primes in Z[i]; cf. art. 8 for the meaning of conjugate); the Gaussian primes in Z, of the form p = 3 (mod 4) = -1 (mod 4), remain
inert (prime in Z[i] as well, reflecting the impossibility of their
being written as the sum of two squares); these yielded a succinct
development of the theorems of Girard and Fermat.
N. B.: C. F. Gauss in his memoir of 1831, and also C. Wessel in 1799 and J.-R. Argand in 1806, 1814, introduced the geometrical interpretation of the complex numbers a+bi = r e^{i \theta} in the complex plane; where a+bi is the Cartesian form of the complex number with abscissa a and ordinate b; and r e^{i \theta} is the polar form of the complex number with magnitude (radius, modulus, absolute value. norm) r and argument (polar angle, phase, azimuth) \theta, and e^{i \theta} = (\cos \theta + i \sin \theta) is the direction factor/direction coëfficient/reduced form/polar arc.
N. B.: The term lateral unit is due to C. F. Gauss, 1831; and the term imaginary unit, due to Renatus Cartesius, 1637. The notation i for the lateral unit is due to L. Euler, 1777.
N. B.: The formula for the direction factor/direction coëfficient/reduced form/polar arc:
e^{i \theta} = (\cos \theta + i \sin \theta)
was obtained by L. Euler in his `Introductio in anaysin infinitorum', 1748; for \theta := \pi it yields:
e^{i \pi} + 1 = 0.
N. B.: The complex numbers were discovered by Girolamo Fazio Cardano (1501-1576) in his `Ars magna', 1545. They were called complex numbers by C. F. Gauss in 1831.
7. The even prime, 2, a prime in Z, splits in Z[i], as 2 = (1+i) (1-i) = (-1+i) (-1-i), and is thus composite in Z[i]; 1+i, 1-i, -1+i, -i-i are the 4 Gaussian prime factors of 2 in Z[i]; these factorisations correspond to the sum of squares form 2 = 1^2 + 1^2.
The first Pythagorean prime, 5, a prime in Z, splits in Z[i], as 5 = (1+2i) (1-2i) = (2+i) (2-i), and is thus composite in Z[i]; 1+2i, 1−2i, −1+2i, −1−2i, 2+i, 2−i, −2+i, −2−i are the 8 Gaussian prime factors of 5 in Z[i]; these factorisations correspond to the sum of squares form 5 = 1^2 + 2^2. cf. the following art. 8 for the eightfold associates of 1+2i, 1-2i.
The first few (<= 1729) Pythagorean primes, i. e., of the form p = 1 (mod 4) are:
5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617, 641, 653, 661, 673, 677, 701, 709, 733, 757, 761, 769, 773, 797, 809, 821, 829, 853, 857, 877, 881, 929, 937, 941, 953, 977, 997, 1009, 1013, 1021, 1033, 1049, 1061, 1069, 1093, 1097, 1109, 1117, 1129, 1153, 1181, 1193, 1201, 1213, 1217, 1229, 1237, 1249, 1277, 1289, 1297, 1301, 1321, 1361, 1373, 1381, 1409, 1429, 1433, 1453, 1481, 1489, 1493, 1549, 1553, 1597, 1601, 1609, 1613, 1621, 1637, 1657, 1669, 1693, 1697, 1709, 1721, ...
For the Pythagorean primes, the sum of squares representation is unique; and they have 8 Gaussian prime factors in Z[i], all corresponding to the sum of squares representation.
The first few (<= 1729) Gaussian primes in Z, i. e., of the form p = 3 (mod 4) = -1 (mod 4), are:
3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131,
139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271,
283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463,
467, 479, 487, 491, 499, 503, 523, 547, 563, 571, 587, 599, 607, 619, 631, 643, 647, 659, 683, 691, 719, 727, 739, 743, 751, 787, 811, 823, 827, 839, 859, 863, 883, 887, 907, 911, 919, 947, 967, 971, 983, 991, 1019, 1031, 1039, 1051, 1063, 1087, 1091, 1103, 1123, 1151, 1163, 1171, 1187, 1223, 1231, 1259, 1279, 1283, 1291, 1303, 1307, 1319, 1327, 1367, 1399, 1423, 1427, 1439, 1447, 1451, 1459, 1471, 1483, 1487, 1499, 1511, 1523, 1531, 1543, 1559, 1567, 1571, 1579, 1583, 1607, 1619, 1627, 1663, 1667, 1699, 1723, ...
For the Gaussian primes p in Z, there are no sum of squares representations; and they have 4 Gaussian prime factors in Z[i], to wit, p, pi, -p, -pi (the associates of p, cf. art. 10).
These threefold kinds of Gaussian primes in Z[i], which are factors of the three kinds of primes in Z, are all of the Gaussian primes (cf. art. 11 for a demonstration).
8. Following C. F. Gauss, call:
i. ||a + bi|| := a^2 + b^2, the norm-squared;
ii. |a + bi| := \sqrt{||a+bi||} = \sqrt{a^2 + b^2}, the norm;
of the complex number a + bi.
N. B.: If \conj (a+bi) := a-bi, called the conjugate of the complex number a+bi, then:
a. \conj{z_1 z_2} = \conj{z_1} \ conj{z_2} (Multiplicativity).
b. ||z|| = z \conj{z}.
c. \conj{z} is the reflection of z about the real axis/x- axis, negating the ordinate. i \conj{z} is the reflection of z about the diagonal line having equation y = x of slope \pi/4.
d. For a complex number z = a + bi, consider the eightfold associates of z and \conj{z}:
z, iz, -z, -iz, \conj{z}, i\conj{z}, -\conj{z}, -i\conj{z}.
which are, respectively, =
a + bi, -b + ai, -a - bi, b - ai, a - bi, b + ai, -a + bi, -b - ai.
These are the possible complex numbers that can be obtained from z by negating either the abscissa or ordinate or by transposing/exchanging them; they are the associates of either z or \conj{z} (cf. art. 10).
e. A complex number z is, together with \conj{z}, a zero of the polynomial:
(x - z) (x - \conj{z}) = x^2 - 2\Re(z) + ||z||
It can be further deduced, for the Gaussian integers and their norms-squared:
i. The norm-squared of a Gaussian integer is a whole number.
ii. For Gaussian integers z_1, z_2:
a. ||1|| = ||i|| = ||-1|| = ||-i|| = 1.
b. ||z_1 z_2|| = (z_1 z_2) \conj {z_1 z_2}
= z_1 z_2 \conj{z_1} \conj{z_2}
= z_1 \conj{z_1} z_2 \conj{z_2}
= ||z_1|| ||z_2|| (Multiplicativity).
9. The Diophantus identity (art. 5) can then be demonstrated as:
(a^2 + b^2) (c^2 + d^2) = ||a + bi|| ||c + di||
= ||(ac-bd) + (ad+bc)i||
= (ac-bd)^2 + (ad+bc)^2
(a^2 + b^2) (c^2 + d^2) = (a^2 + b^2) (d^2 + c^2)
= ||a + bi|| ||d + ci||
= ||(ad-bc) + (ac+bd)i||
= (ad-bc)^2 + (ac+bd)^2
= (ac+bd)^2 + (ad-bc)^2
to yield:
(a^2 + b^2) (c^2 + d^2) = e^2 + f^2
where:
either: e = ac-bd; f = ad+bc
or: e = ac+bd; f = ad-bc.
10. Call i. -1. -i. 1, the units of the ring of Gaussian integers Z[i]; and for a Gaussian integer z; z, iz, -z, -iz, its associates in Z[i], obtained by multiplying by the units, rotating by right angles through the complex plane.
Likewise, call 1, -1, the units of the ring of integers Z; and for an integer m; m, -m, its associates in Z, obtained by multiplying by the units, rotating by straight angles through the complex plane.
N. B.: The units are the only elements of the ring with norm 1, i. e., lying on the unit circle; and with their reciprocals also in the ring.
N. B.: The units of Z[i] are the biquadratic {quartic} roots of unity, i. e.:
\radical{1}{4} = 1^{1/4} = i. -1. -i. 1 = e^{i \pi/2}, e^{i 2\pi/2}, e^{i 3\pi/2}, e^{i 4\pi/2}
Likewise, the units of Z are the square roots of unity, i. e.:
\sqrt{1} = \radical {1}{2} = 1^{1/2} = -1. 1 = e^{i \pi}, e^{i 2\pi}
The nth roots of unity, i. e., \radical{1}{n} = 1^{1/n}, are the n zeros of the polynomial:
x^n - 1 = (x-1) (x^{n-1} + x^{n-2} + ... + x + 1)
These are formed by rotating \omega := e^{i 2\pi/n} through the complex plane, i. e.:
\omega_k := \omega^k = e^{i 2\pi/n k} for k in [0..n)
i. e.:
\omega = \omega_1 = e^{i 2\pi/n}
generates the nth roots of unity.
A primitive nth root of unity is any such root that generates the n roots; these are:
\omega_k := \omega^k = e^{i 2\pi/n k} for k in [0..n) with k relatively prime to n.
The nth roots of unity are said to be conjugates with respect to the polynomial x^n-1; the conjugates of 1, to wit, \omega_k for k in [1..n), are the zeros of the polynomial:
x^{n-1} + x^{n-2} + ... + x + 1 = \prod_{k in [1..n)} (x - \omega_k).
For n = 2, \omega_1 = -1 is the primitive square root of unity; and for n = 4, \omega_1 = i and \omega_3 = -i are the primitive biquadratic {quartic} roots of unity.
For n=2, \omega = \omega_1 = e^{i \pi} = -1 is a zero of x+1, yielding:
e^{i \pi} + 1 = 0.
11. A Gaussian prime (in Z) is also a Gaussian prime in Z[i] (as shown in art. 2).
A Pythagorean prime p is the norm-squared of a Gaussian integer z in Z[i], i. e., p = ||z||.
For Gaussian primes z = r+si in Z[i], either z is real (i. e., \Im(z)=s=0, and z is a Gaussian prime in Z); or z is non-real (i. e., has non-zero ordinate/lateral part/imaginary part), and n = ||z|| = r^2+s^2, a natural number that can be written as a sum of two squares; if n = ||z|| is not prime, its proper non-unit factors, say f_1, f_2, with n = f_1 f_2 would yield f_1 f_2 = (r + si) (r - si), and, due to unique factorisation in the Gaussian integers, proper non-unit factors for z in the Gaussian integers, an absurdity. Therefore, n = ||z|| = r^2+s^2 is a Pythagorean prime (or, if z = 1+i, the even prime, 2).
For Pythagorean primes p, let p = r^2 + s^2 = ||r + si|| = ||z||, where z=r+si is a Gaussian integer; if z is not a Gaussian prime in Z[i], then, its proper non-unit factors, say z_1, z_2, with z = z_1 z_2 would yield p = ||z_1|| ||z_2||, proper non-unit factors for p, an absurdity. Therefore, z=r+si is a Gaussian prime (in Z[i]).
For the even prime p = 2 = ||1+i||, z=1+i is a Gaussian prime (in Z[i]).
Therefore, the primes in the Gaussian integers Z[i] are (as enunciated in art. 7):
i. 1+i (the proper non-unit factor of 2; and its associates in Z[i]).
ii. Proper non-unit factors of the Pythagorean primes (and their associates in Z[i]).
iii. Gaussian primes in Z (and its associates in Z[i]).
N. B.: 2 is a perfect square up to units in Z[i], since its proper factor 1+i has its conjugate 1-i being simultaneously its associate.
N. B.: Purely lateral/imaginary perfect squares are of the form 2a^2 i = (a + ai)^2; the smallest such is 2i = (1+i)^2.
These primes can be found on concentric circles emanating from the origin in the complex plane, having radii/norms: \sqrt{2} (case i), \sqrt{p} where p is a Pythagorean prime (case ii), and p where p is a Gaussian prime (case iii).
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