1. On 25 Dec. 1640, P. de Fermat (1607-1665), of Tolosa {Toulouse}, Aquitania/Occitania {Aquitaine/Occitanie}, Provincia Romana/Provincia nostra/Gallia Narbonensis/Gallia Ulterior/Gallia Trans-Alpina, W. Francia, composed and despatched a missive to M. Mersenne, OM, about the two squares theorem for prime numbers, called Fermat's Christmas theorem.
Only some numbers can be written as a 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 the letter that among (odd) primes, those (and, obviously, as shown below (art. 2, proposition III), 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.
Definition: (Odd) primes of the form p = 1 (mod 4) are called Pythagorean primes (since the sum of two squares is reminiscent of the Pythagoras theorem); in contradistinction, (odd) primes of the form p = 3 (mod 4) = -1 (mod 4) are called Gaussian primes (the reason for the name is explained in art. 6 below).
Proposition I (P. de Fermat): The Pythagorean primes can be written as a sum of two squares.
Unlike the Pythagorean prime 5; the Gaussian primes 3, 7, or 11, cannot be so written; the next Pythagorean prime 13 = 2^2 + 3^2.
Note bene: The even prime 2 = 1^2 + 1^2.
Proposition II: If a prime is written as a sum of two numbers, they, the addends, are relatively prime (provided that the addends are proper, i. e., less than the sum).
2. Demonstration: For (odd) primes that can be so written, it is clear that they must be written as the sum of an odd square and an even square, i. e., squares of different parities.
Odd squares are of the form x = 1^2, 3^2/(-1)^2 = 1 (mod 4); Even squares, of the form y = 0^2, 2^2 = 0 (mod 4). Thus their sum is of the form z = 1 (mod 4). Q. E. D. Therefore:
Proposition III: If a prime can be written as a sum of two squares, then it is either even or a Pythagorean prime. The Gaussian primes cannot be so written.
N. B.: This is the converse of the Christmas theorem (proposition I).
What is not evident 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', 1798, 1801.
4. A. Girard (1595-1632) had stated, in 1625, with no demonstrations, that:
Proposition IV (A. Girard): The numbers that can be written as a sum of two squares are:
i. The even prime.
ii. Squares. (N. B.: 0^2 is also permissible in the sum.)
iii. The Pythagorean primes.
iv. Product of such numbers.
(in his commentary and annotated edition of `L’arithmetique', 1585, of S. Stevinus).
5. Diophantus of Alexandria (fl. IIIrd century) showed (in his `Arithmetica') that:
Proposition V (Diophantus of Alexandria): If two numbers can be written as sums of two squares, then so can be their product (and, in two different ways).
Demonstration:
(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.
Q. E. D.
Nota bene: 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. 10, Proposition IX).
6. C. F. Gauss in his second memoir on biquadratic {quartic} reciprocity,
`Theoria residuorum biquadraticorum', 1831, discovered the ring Z[i] of
Gaussian integers/Gaussian numbers 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. 11 on the lateral unit), and showed that
unique factorisation into primes (the fundamental theorem of arithmetic) holds for the Gaussian integers/Gaussian numbers.
The writing of a number as a sum of two squares can be reïnterpreted as
splitting it into conjugate Gaussian integers (cf. art. 9 for the meaning of conjugate).
Some primes in Z split/decompose/factorise into conjugate
primes in the Gaussian integers, the ring Z[i], reflecting their being written as a sum of
two squares (cf. art. 12 for a demonstration that these factors are primes in Z[i], Proposition XI).
The other primes in Z remain
inert, prime in the ring Z[i] as well, reflecting the impossibility of their
being written as a sum of two squares.
Definition: Call the primes (in Z) that split into conjugate primes in the Gaussian integers Z[i], the splitting primes; and the primes (in Z) that remain inert in the Gaussian integers Z[i], the inert primes. In art. 2, it was shown (proposition III, reënunciated here):
Proposition VI: A splitting prime is either even or a Pythagorean prime. The Gaussian primes are inert.
Fermat and Girard enunciated that:
Proposition VII (P. de Fermat): The Pythagorean primes split. The inert primes are Gaussian.
7. 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}
(with a, b, r, \theta, all real numbers, and i, the lateral unit, cf. below) in the complex plane; where:
a+bi
is the Cartesian form of the complex number with abscissa/real part a and ordinate/lateral
part/imaginary part b; and:
r e^{i \theta}
is the polar form of the complex number with magnitude (radius, modulus, absolute value, metric, measure, size) 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 factor/polar coëfficient/angular factor/angular coëfficient/phase factor/phase coëfficient/azimuthal factor/azimuthal coëfficient/polar arc.
Definition: Call i = \sqrt{-1} the lateral unit or imaginary unit.
Call a complex number z = a+bi:
1. real if \Im(z):=b=0.
2. non-real if it has non-zero ordinate/lateral
part/imaginary part, i. e., \Im(z)=b \neq 0.
3. lateral/imaginary if \Re(z):=a=0.
4. non-lateral/non-imaginary if it has non-zero abscissa/real part, i. e., \Re(z)=a \neq 0.
5. axial, if it is either real or lateral/imaginary.
6. non-axial, if it is neither real nor lateral/imaginary.
(End of definition.)
Nota bene: There are two square roots of unity, to wit, \sqrt{-1} and -\sqrt{-1}; these are indistinguishable and which one is assigned to the lateral unit i is of no concern (cf. also art. 11 on primitive square roots and primitive biquadratic {quartic} roots of unity).
Nota bene: 0 is the only complex number that is both real and lateral/imaginary.
The complex numbers (C=R[i]) were discovered by Girolamo Fazio Cardano {Hieronymus Photius Cardanus} (1501-1576) in his `Ars magna' {`Artis magnae'}, 1545. They were called complex numbers by C. F. Gauss in 1831.
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 in his lecture, `De
formulis differentialibus angularibus maxime irrationalibus quas tamen
per logarithmos et arcus circulares integrare licet' [E671], Academiae
imperialis scientiarum, Petropolis {St. Petersburg}, Russia, May 1777
(5 days after the birth of C. F. Gauss); the next use of the notation is
by C. F. Gauss in his `Disquisitiones arithmeticae', 1798, 1801.
1777
30 Apr. -> Birth of C. F. Gauss, Braunschweig {Brunsvici}, Fürstentum Braunschweig-Wolfenbüttel {Principatus Brunsvicensis et Guelpherbytensis}, Germania {Heiliges Römisches Reich Deutscher nation / Imperium Sacrum Romanum nationis Germanicae}.
5 May -> L. Euler introduces the notation i = \sqrt{-1}, Petropolis, Russia.
The formula for the polar coëfficient/reduced form:
e^{i \theta} = (\cos \theta + i \sin \theta)
was obtained by L. Euler in his `Introductio in analysin infinitorum', 1745, 1748; for \theta := \pi it yields:
e^{i \pi} + 1 = 0.
8. 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 non-axial 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 non-axial 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 (<= 1777) 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, 1733, 1741, 1753, 1777 ...
For the splitting primes, the sum of squares representation is unique; and they have 8 non-axial prime factors in Z[i], all corresponding to the sum of squares representation.
The first few (<= 1777) 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, 1747, 1759 ...
For the inert primes, there are no sum of squares representations; and they have 4 axial prime factors in Z[i], to wit, p, pi, -p, -pi (the associates of p, cf. art. 11).
Proposition VIII: The threefold kinds of primes in Z[i], which are factors of the three kinds of primes in Z; to wit: even (also splitting), splitting (and odd), and inert; are all of the primes in Z[i](cf. art. 12 for a demonstration, of Proposition XII).
9. Definition: Following C. F. Gauss, call:
i. ||a + bi|| := a^2 + b^2 = r^2, the norm;
ii. |a + bi| := \sqrt{||a+bi||} = \sqrt{a^2 + b^2} = r, the metric/magnitude/measure/modulus/radius/absolute value/size;
of the complex number a + bi = r e^{i \theta}.
Nota bene: Call \conj{a+bi} := a-bi = r e^{-i \theta} the conjugate of the complex number a+bi; then:
a. The conjugate of a product is the product of the conjugates; i. e.:
\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 (and the polar angle). 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 = r e^{i \theta}, 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, =
r e^{i \theta}, r e^{i (\theta + \pi/2)}, r e^{i (\theta + \pi)}, r e^{i (\theta + 3\pi/2)},
r e^{-i \theta}, r e^{i (-\theta + \pi/2)}, r e^{i (-\theta + \pi)}, r e^{i (-\theta + 3\pi/2)}.
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. 11).
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:
i. The norm of a Gaussian integer is a whole number.
ii.
a. The norm of a unit (cf. art. 11) is 1; i. e.:
||1|| = ||i|| = ||-1|| = ||-i|| = 1
b. The norm of a product is the product of the norms;
i. e., for Gaussian integers z_1, z_2:
||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).
iii. The norm of a Gaussian integer is a sum of two squares.
10. The writing of a number as a sum of two squares can be reïnterpreted as the writing it as the norm of a Gaussian integer.
The Diophantus identity (art. 5, proposition V) can be reënunciated as:
Proposition IX (Diophantus of Alexandria): If two numbers X = ||a+bi||, Y = ||c+di||, can be written as norms of Gaussian integers, then so can be their product Z = ||e+fi|| (and, in two different ways).
Demonstration:
XY = (a^2 + b^2) (c^2 + d^2)
= ||a + bi|| ||c + di||
= ||(ac-bd) + (ad+bc)i|| (Multiplicativity)
= (ac-bd)^2 + (ad+bc)^2
XY = (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|| (Multiplicativity)
= (ad-bc)^2 + (ac+bd)^2
= (ac+bd)^2 + (ad-bc)^2
yielding:
XY = ||a + bi|| ||c + di|| = ||e + fi|| = Z
where:
either: e = ac-bd; f = ad+bc
or: e = ac+bd; f = ad-bc
and Z = ||e+fi||.
Q. E. D.
11. 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.
Nota bene: 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.
Nota bene: 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; these 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.
Nota bene: Which of the units of Z is assigned ex nihilo/ab initio to be the primitive square root of unity in C=R[i] (equivalently, in Z[i]) is of no concern, provided it is consistent with the assignment of the primitive biquadratic {quartic} root of unity in Z[i] (cf. art. 7 on assigning \sqrt{-1} to the lateral unit i).
For n=2, \omega = \omega_1 = e^{i \pi} = -1 is a zero of x+1, yielding:
e^{i \pi} + 1 = 0 (as obtained in art. 7 too).
12. An inert prime is a prime in Z[i]; the other primes in Z[i] are as follows.
Demonstration: For primes z = r+si in Z[i], either z is real/axial and is an inert prime; or z is non-axial 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 n_1, n_2, with n = n_1 n_2 would yield n_1 n_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 splitting prime. Q. E. D. Hence:
Proposition X: A prime in Z[i] is either an inert prime (and real/axial) or it is non-axial with its norm a splitting prime.
A splitting prime p is the norm of a non-axial Gaussian integer z in Z[i], i. e., p = ||z||; the question arises whether these factors are primes or composites in Z[i].
Demonstration: For splitting primes p, let p = r^2 + s^2 = ||r + si|| = ||z||, where z=r+si is a non-axial Gaussian integer; if z is not a 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 prime in Z[i]. Q. E. D.
Definition: Call the two non-axial proper non-unit factors in Z[i] of an odd splitting prime, the Pythagorean primes in Z[i].
Hence:
Proposition XI: The factors of an odd splitting prime are conjugate non-axial Pythagorean primes in Z[i].
Therefore (reënunciating proposition VIII in art. 8):
Proposition XII: The primes in the Gaussian integers Z[i] are:
i. 1+i (the proper non-unit factor of 2; non-axial); and its associates in Z[i].
ii. Pythagorean primes in Z[i], i. e., non-axial proper non-unit factors of the odd splitting primes; and their associates in Z[i].
iii. Inert primes (real/axial); and their associates in Z[i].
Nota bene: 2 is a perfect square up to units in Z[i], since its proper non-unit factor 1+i has its conjugate 1-i being simultaneously its associate.
Nota bene: The non-axial proper non-unit factor of 2, 1+i, is neither a Gaussian prime (in
Z) nor a Pythagorean prime in Z[i], just as 2 is neither a Gaussian nor a Pythagorean prime in Z.
Nota bene: Lateral/imaginary perfect squares are of the form 2a^2 i = (a + ai)^2 = a^2(1+i)^2; the smallest such is 2i = (1+i)^2.
Nota bene: Lateral/imaginary primes in Z[i] are associates of inert primes in Z.
These primes can be found on concentric circles emanating from the origin in the complex plane, having radii/moduli/metrics/magnitudes/measures:
i. \sqrt{2}. (i. e., |1+i|)
ii. \sqrt{p} where p is an odd splitting prime.
(i. e., if r+si is the non-axial Pythagorean prime factor of p, then |r+si|, which happens to be also \sqrt{|p|}.)
iii. p (i. e., |p|), an inert prime.
and norms:
i. 2. (i. e., ||1+i||)
ii. p where p is an odd splitting prime.
(i. e., if r+si is the non-axial Pythagorean prime factor of p, then ||r+si||, which happens to be also |p|.)
iii. p^2 (i. e., ||p||), where p is an inert prime.
13. Reïnterpreting the writing of a number as a sum of two squares as
the writing it as the norm of a Gaussian integer:
Demonstration: Consider a natural number n that can be written as the norm of a Gaussian integer z, i. e., n = ||z||, the prime factorisation of n, and the corresponding prime factorisation of z (obtained by taking one of the conjugate Pythagorean prime factors in the case of the splitting prime factors of n). From the multiplicativity of the norm, it follows that n is the product of the three types of norms enumerated in the preceding art. 12, wherein all the primes in Z[i] contribute for each instance a prime of single multiplicity to the product n, except that the inert primes contribute primes of double multiplicity. Q. E. D. Therefore:
Proposition XIII: If a number can be written as the norm of a Gaussian integer, then all the inert primes in its prime factorisation will be of even multiplicity.
and an analogue of Girard's theorem (art. 4, proposition IV) follows at once:
Proposition XIV (A. Girard): The numbers that can be written as the norm of a Gaussian integer are:
i. The even prime.
ii. Square. (N. B.: 0^2 is also permissible in the sum.)
iii. A splitting prime.
iv. Product of such numbers.
e. g., consider the numbers 1640, 1777, 1545, 1637, 1745, 1748, 1773, 1798, 1801, and 1831, and whether they can be written as the norms of some Gaussian integers:
1640, can be factorised by considering numbers up to \floor (\sqrt{1640}) = 40; this yields:
1640 = 41 * 5 * 2^3
with:
41 = 5^2 + 4^2 = ||5+4i||
5 = 2^2 + 1^2 = ||2+i||
2^3 = 2^2(1^2 + 1^2) = 2^2 + 2^2 = 2^2 ||1+i||
i. e., each of the odd prime factors 41, 5, splitting in Z[i] into Pythagorean primes a+bi = 5+4i, c+di=2+i, respectively, and yielding, through Diophantus's identity:
||5+4i|| ||2 \pm i|| = ||6 + 13i|| or ||14 + 3i||
i. e., either e+fi = 6 + 13i or e+fi = 14 + 3i, and then on taking into account 2^2 ||1 \pm i||, yielding:
1640 = 2^2 ||19 + 7i|| = 2^2 ||17 + 11i||.
1777, can, by considering numbers up to \floor (\sqrt{1777}) = 42 (N. B.: 42^2 = 1764), be seen to be a (Pythagorean) prime; it can be seen to be a splitting prime, by considering squares of numbers up to \floor (\sqrt{1777/2}) = \floor (\sqrt{888}) = 29 (N. B.: 29^2 = 841); its Pythagorean prime factor is 39 + 16i; and it splits as:
1777 = ||39+16i||.
1545, can be factorised by considering numbers up to \floor (\sqrt{1545}) = 39 (N. B.: 39^2 = 1521); this yields:
1545 = 103 * 5 * 3
where 103 and 3 are Gaussian primes (thus, inert) occurring with odd multiplicity; therefore 1545 cannot be so written.
1637, can, by considering numbers up to \floor (\sqrt{1637}) = 40, be seen to be a (Pythagorean) prime; it can be seen to be a
splitting prime, by considering squares of numbers up to \floor
(\sqrt{1637/2}) = \floor (\sqrt{816}) = 28 (N. B.: 28^2 = 784); its
Pythagorean prime factor is 31 + 26i; and it splits as:
1637 = ||31+26i||.
1745, can be factorised by considering numbers up to \floor (\sqrt{1745}) = 41 (N. B.: 41^2 = 1681); this yields:
1745 = 349 * 5
where 349, a (Pythagorean) prime, can be seen to be a
splitting prime, by considering squares of numbers up to \floor
(\sqrt{349/2}) = \floor (\sqrt{174}) = 13 (N. B.: 13^2 = 169); its
Pythagorean prime factor is 18 + 5i; and it splits as:
349 = ||18 + 5i||
and:
5 = 2^2 + 1^2 = ||2+i||
i. e., each of the odd prime factors 349, 5, splitting in Z[i] into
Pythagorean primes a+bi = 18+5i, c+di=2+i, respectively, and yielding,
through Diophantus's identity:
||18+5i|| ||2 \pm i||| = ||41 + 8i|| or ||31 + 28i||
i. e., either e+fi = 41 + 8i or e+fi = 31 + 28i, yielding:
1745 = ||41 + 8i|| = ||31 + 28i||.
1748, can be factorised by considering numbers up to \floor (\sqrt{1748}) = 41 (N. B.: 41^2 = 1681); this yields:
1748 = 23 * 19 * 2^2
where 23 and 19 are Gaussian primes (thus, inert) occurring with odd multiplicity; therefore 1748 cannot be so written.
1773, can be factorised by considering numbers up to \floor (\sqrt{1773}) = 42 (N. B.: 42^2 = 1764); this yields:
1773 = 197* 3^2
where 3 is a Gaussian prime (thus, inert) occurring with even multiplicity, and 197, a (Pythagorean) prime, can be seen to be a
splitting prime, by considering squares of numbers up to \floor
(\sqrt{197/2}) = \floor (\sqrt{98}) = 9 (N. B.: 9^2 = 81); its
Pythagorean prime factor is 14 + i; and it splits as:
197 = ||14 + i||
and thus:
1773 = 3^2 ||14 + i||.
1798, can be factorised by considering numbers up to \floor (\sqrt{1798/2}) = \floor (\sqrt{899}) = 29 (N. B.: 29^2 = 841); this yields:
1798 = 31 * 29 * 2
where 31 is a Gaussian prime (thus, inert) occurring with odd multiplicity; therefore 1798 cannot be so written.
1801, can, by considering numbers up to \floor (\sqrt{1801}) = 42 (N. B.: 42^2 = 1764), be seen to be a (Pythagorean) prime; it can be seen to be a
splitting prime, by considering squares of numbers up to \floor
(\sqrt{1801/2}) = \floor (\sqrt{900}) = 30; its
Pythagorean prime factor is 35 + 24i; and it splits as:
1801 = ||35+24i||.
1831, can, by considering numbers up to \floor (\sqrt{1831}) = 42 (N. B.: 42^2 = 1764), be seen to be a Gaussian prime, and therefore, inert; therefore 1831 cannot be so written.
The whole numbers up to 101 that can be so written are:
0 = ||0||.
1 = ||1||.
2 = ||1+i||. Even prime.
4 = 2^2.
5 = ||2+i||. Pythagorean prime.
8 = 2^2 ||1+i||.
9 = ||3||. 3^2.
10 = ||3+i||. 5 * 2.
13 = ||3+2i||. Pythagorean prime.
16 = 2^4.
17 = ||4+i||. Pythagorean prime.
18 = 3^2 ||1+i||. 3^2 * 2.
20 = 2^2 ||2+i||. 5 * 2^2.
25 = ||5|| = ||4+3i||. 5^2.
26 = ||5+i||. 13 * 2.
29 = ||5+2i||. Pythagorean prime.
32 = 2^4 ||1+i||.
34 = ||5+3i||. 17 * 2.
36 = ||6||. 3^2 * 2^2.
37 = ||6+i||. Pythagorean prime.
40 = 2^2 ||3+i||. 5 * 2^3.
41 = ||5+4i||. Pythagorean prime.
45 = 3^2 ||2+i||. 5 * 3^2.
49 = ||7||. 7^2.
50 = ||7+i|| = 5^2 ||1+i||. 5^2 * 2.
52 = 2^2 ||3+2i||. 13 * 2^2.
53 = ||7+2i||. Pythagorean prime.
58 = ||7+3i||. 29 * 2.
61 = ||6+5i||. Pythagorean prime.
64 = 2^6.
65 = ||8+i|| = ||7+4i||. 13 * 5.
68 = 2^2 ||4+i||. 17 * 2^2.
72 = 6^2 ||1+i||. 3^2 * 2^3.
73 = ||8+3i||. Pythagorean prime.
74 = ||7+5i||. 37 * 2.
80 = 2^4 ||2+i||. 5 * 2^4.
81 = ||9||. 3^4.
82 = ||9+i||. 41 * 2.
85 = ||9+2i|| = ||7+6i||. 17 * 5.
89 = ||8+5i||. Pythagorean prime.
90 = 3^2 ||3+i||. 5 * 3^2 * 2.
97 = ||9+4i||. Pythagorean prime.
98 = 7^2 ||1+i||. 7^2 * 2.
100 = ||10|| = 2^2 ||4+3i||. 5^2 * 2^2.
101 = ||10+i||. Pythagorean prime.
