# References — Chapter ᚔ(I): The first landing *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-21, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch21.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 21.1 {#r-21.1} Euler’s formula is e^(iθ) = cos θ + i sin θ, and θ = π gives Euler’s identity e^(iπ) + 1 = 0; it was published in Euler’s Introductio in analysin infinitorum (1748); Roger Cotes had given the equivalent logarithmic form in 1714. Multiplying by a number of modulus 1 adds its argument and leaves the modulus unchanged. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §4.21](https://dlmf.nist.gov/4.21) — gives the complex trigonometric identities and de Moivre theorem; the θ = π substitution is direct arithmetic. - [The Euler Archive, E101: Introductio in analysin infinitorum, volume 1](https://scholarlycommons.pacific.edu/euler-works/101/) — identifies Euler’s Introductio in analysin infinitorum, in which he “lays the foundations of modern mathematical analysis,” with publication date 1748, and links a free scan of the text. - [Euler’s formula, Wikipedia](https://en.wikipedia.org/wiki/Euler%27s_formula) — dates the publication of e^(ix) = cos x + i sin x to the Introductio of 1748 and records Roger Cotes’s 1714 Logometria, whose geometrical argument gives ix = ln(cos x + i sin x). - [Complex number, Wikipedia (multiplication in polar form)](https://en.wikipedia.org/wiki/Complex_number) — read as raw wikitext: in polar form “the absolute values are multiplied and the arguments are added to yield the polar form of the product.” ### 21.2 {#r-21.2} Euler’s identity links five fundamental constants and has twice headed a published poll — the Mathematical Intelligencer’s of 1990 and Physics World’s of 2004 — and the Benjamin Peirce quotation comes from notes taken by his student W. E. Byerly. *Status: judgement* - [Euler’s identity, Wikipedia](https://en.wikipedia.org/wiki/Euler%27s_identity) — records the Peirce quotation and attributes it to a Harvard lecture; it also reports descriptions of the identity as famous/beautiful; its bibliography identifies Kasner and Newman (1940) as a source for the quotation, and it records the 1990 Mathematical Intelligencer poll naming the identity the most beautiful theorem in mathematics and the 2004 Physics World poll in which it tied with Maxwell’s equations as the greatest equation ever. - [Benjamin Peirce, Wikiquote](https://en.wikiquote.org/wiki/Benjamin_Peirce) — sources the quotation to notes by W. E. Byerly published in R. C. Archibald, Benjamin Peirce, 1809-1880: Biographical Sketch and Bibliography (1925), and again in Kasner and Newman (1940), and attaches it to Euler’s identity. Editorial note: The card used to say the identity is voted the most beautiful equation whenever anyone holds a vote. No source supports a claim that broad, so we replaced it with the two polls that are actually on the record, 1990 and 2004. The quotation’s chain of transmission is now traced to its earliest printing — Byerly’s notes in Archibald (1925) — but Byerly was writing down a lecture from memory and no verbatim transcript survives, so the exact wording stays a judgement rather than a quotation from the lecturer’s own hand. ### 21.3 {#r-21.3} Complex conjugation sends a + bi to a − bi, and z times its conjugate equals a² + b² = |z|²; hence i times its conjugate is 1 and the modulus of e^(iθ) is 1. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §1.9(i), Complex Numbers](https://dlmf.nist.gov/1.9#i) — the conjugate, “1.9.11 z̄ = x − iy,” and “1.9.12 |z̄| = |z|”; the identities follow by substituting z = a + bi and i (the line formerly pointed to §4.21(iv), which is about trigonometric functions of a complex argument). - [Complex number, Wikipedia](https://en.wikipedia.org/wiki/Complex_number) — confirms conjugation, modulus, and the geometric interpretation of multiplication. ### 21.4 {#r-21.4} The Gaussian integers Z[i] have norm N(a+bi)=a²+b²; N(1+i)=2, and 2 factors as −i(1+i)², so 2 is not prime in Z[i]. Its units, the elements whose inverse is again a Gaussian integer, are 1, −1, i and −i. Primes congruent to 1 modulo 4 split while primes congruent to 3 modulo 4 remain Gaussian prime up to associates. Gauss introduced the ring in his second monograph on quartic (biquadratic) reciprocity, 1832. *Status: confirmed* - [Gaussian integer, Wikipedia](https://en.wikipedia.org/wiki/Gaussian_integer) — confirms the Gaussian-integer norm, its multiplicativity, the factorization of 2, and the split/inert behavior by congruence modulo 4; its opening sentence attributes the ring to Gauss’s second monograph on quartic reciprocity (1832), while its own reference list dates that monograph 1831; “The units of the ring of Gaussian integers (that is the Gaussian integers whose multiplicative inverse is also a Gaussian integer) are precisely the Gaussian integers with norm 1, that is, 1, −1, i and −i.” - [Carl Friedrich Gauss, MacTutor](https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/) — links “An extract from Theoria residuorum biquadraticorum (1828 - 32),” a span consistent with a second memoir printed in 1832; the page does not date the two memoirs separately. ### 21.5 {#r-21.5} For real θ, e^(iθ) lies on the unit circle and multiplication by it rotates without changing modulus. In polar form the moduli multiply and the arguments add. *Status: confirmed* - [Euler’s formula, Wikipedia](https://en.wikipedia.org/wiki/Euler%27s_formula) — gives e^(iθ)=cos θ+i sin θ, says e^(iφ) “traces out the unit circle in the complex plane as φ ranges through the real numbers,” and says “The polar form simplifies the mathematics when used in multiplication or powers of complex numbers”; the polar product rule on the Complex number line below is what makes multiplication by a unit factor a rotation in the complex plane. - [NIST Digital Library of Mathematical Functions, §4.21(iv)](https://dlmf.nist.gov/4.21#iv) — supplies the modulus identities; cos²θ+sin²θ=1 gives the stated value. - [Complex number, Wikipedia (multiplication in polar form)](https://en.wikipedia.org/wiki/Complex_number) — read as raw wikitext: in polar form “the absolute values are multiplied and the arguments are added to yield the polar form of the product.” ### 21.6 {#r-21.6} Split-complex numbers use a unit j with j²=+1 and j≠±1; they are associated with Cockle’s tessarines and have a hyperbola/Lorentz-boost interpretation. They are one of exactly three two-dimensional unital algebras over the reals, the others being the complex numbers (i²=−1) and the dual numbers (ε²=0); Clifford used them as the coefficients of his split-biquaternions in 1873. Within Cockle’s tessarines they are the subalgebra of real tessarines w + yj. Their squared modulus is x² − y² and is multiplicative, and its level sets are hyperbolas, so multiplying by a number of squared modulus 1 keeps each point on its hyperbola. *Status: confirmed* - [Split-complex number, Wikipedia](https://en.wikipedia.org/wiki/Split-complex_number) — confirms the defining relation, the historical association with tessarines (Cockle, 1848), the unit hyperbola, and the sentence ‘Since the late twentieth century, the split-complex multiplication has commonly been seen as a Lorentz boost of a spacetime plane’; “the product of a number z with its conjugate is N(z) := zz∗ = x² − y²,” “It has the composition algebra property: ‖zw‖ = ‖z‖‖w‖,” and “The set of points {z : ‖z‖² = a²} is a hyperbola for every nonzero a.” - [Hypercomplex number, Wikipedia](https://en.wikipedia.org/wiki/Hypercomplex_number) — ‘Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary complex numbers, the split-complex numbers, and the dual numbers’, and notes that every such algebra is associative and commutative. - [Split-biquaternion, Wikipedia](https://en.wikipedia.org/wiki/Split-biquaternion) — ‘This algebra was introduced by William Kingdon Clifford in an 1873 article for the London Mathematical Society’, with split-complex coefficients. - [Bicomplex number, Wikipedia](https://en.wikipedia.org/wiki/Bicomplex_number) — read as raw wikitext: “The tessarines are now best known for their subalgebra of real tessarines t = w + y j, also called split-complex numbers, which express the parametrization of the unit hyperbola.” ### 21.7 {#r-21.7} Dual numbers have ε²=0 with ε≠0; their first-order algebra gives f(x+ε)=f(x)+f′(x)ε, which underlies forward-mode automatic differentiation. William Clifford introduced them in 1873. *Status: confirmed* - [Dual number, Wikipedia](https://en.wikipedia.org/wiki/Dual_number) — confirms the dual-number definition, nilpotent ε, the derivative expansion, and use in automatic differentiation, and opens ‘Dual numbers were introduced in 1873 by William Clifford’. ### 21.8 {#r-21.8} The complex plane identifies a+bi with (a, b); Caspar Wessel first described a complex number as a point of the plane, in a memoir presented to the Copenhagen Academy in 1797 and published in 1799; Jean-Robert Argand independently issued a pamphlet in 1806, with a further paper in 1814; Gauss is dated at 21.4, which rules on 1831 for the announcement and 1832 for the printing; and complex arithmetic cannot be given an order compatible with field arithmetic, though it keeps the associative, commutative and distributive laws and division by nonzero numbers. Wessel’s paper was presented to the Royal Danish Academy in 1797; Gauss’s announcement of the second memoir appeared in the Göttingische gelehrte Anzeigen of 23 April 1831. Gauss’s 1831 work largely established modern notation and terminology, the term complex number included. *Status: judgement* - [Complex number, Wikipedia](https://en.wikipedia.org/wiki/Complex_number) — confirms the coordinate-plane model, the Wessel/Argand/Gauss history, and the non-orderability of the complex field. Re-read raw on 17 September 2026 for the years: “The idea of a complex number as a point in the complex plane was first described by Danish–Norwegian mathematician Caspar Wessel in 1799 … Wessel’s memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers,” and the page quotes the Encyclopedia of Mathematics: “The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the ‘Argand diagram’, came into use after the publication in 1806 and 1814 of papers by J. R. Argand”; of Gauss: “It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology,” and he “introduced the term complex number for a + bi”; the complex numbers support “the same basic arithmetic operations of addition, subtraction, multiplication, and division, satisfying the same associative, commutative, and distributive laws,” and “Unlike the reals, C is not an ordered field … in any ordered field, the square of any element is necessarily positive, so i² = −1 precludes the existence of an ordering on C.” - [Euler’s formula, Wikipedia](https://en.wikipedia.org/wiki/Euler%27s_formula) — confirms the historical description of Wessel’s complex-plane representation. - [Caspar Wessel, Wikipedia](https://en.wikipedia.org/wiki/Caspar_Wessel) — read as raw wikitext: “His fundamental paper, Om directionens analytiske betegning (On the Analytical Representation of Direction), was presented in 1797 to the Royal Danish Academy of Sciences and Letters.” - [Deutsches Textarchiv, TEI header of Gauß, Anzeige von Theoria residuorum biquadraticorum, commentatio secunda](https://www.deutschestextarchiv.de/api/tei_header/gauss_theoria_1831) — the archive’s record, served as plain XML: “Göttingische gelehrte Anzeigen, 23. April 1831.” Editorial note: The broad historical sequence is supported, but exact priority language such as ‘made respectable’ and ‘fixed the modern view’ is interpretive, so we record it as a judgement call. ### 21.9 {#r-21.9} De Moivre’s formula (cos θ+i sin θ)^n=cos(nθ)+i sin(nθ) gives (cos(π/3)+i sin(π/3))^6=1 and describes the sixth roots of unity on a regular hexagon. De Moivre worked with the relation from 1707 onward, including in the Miscellanea Analytica of 1730. Sources differ on whether he wrote it in this form: MacTutor says it appears so in his paper of 1722, the De Moivre’s formula article says he never stated it; corrections welcome. It follows from Euler’s formula and the exponential law. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §4.21, De Moivre’s Theorem](https://dlmf.nist.gov/4.21) — equation 4.21.34 states de Moivre’s theorem for integer n; substituting θ=π/3 and n=6 gives cos 2π+i sin 2π=1. - [Roots of unity, Wikipedia](https://en.wikipedia.org/wiki/Root_of_unity) — confirms the geometric regular-polygon description of roots of unity. - [De Moivre’s formula, Wikipedia](https://en.wikipedia.org/wiki/De_Moivre%27s_formula) — names the formula for de Moivre ‘although he never stated it in his works’, cites his 1707 paper and the Miscellanea Analytica of 1730, where he considers cos θ and cos nθ explicitly, and derives the formula from Euler’s. - [Abraham de Moivre, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/De_Moivre/) — on the formula for (cos x + i sin x)ⁿ: “It appears in this form in a paper which de Moivre published in 1722, but a closely related formula had appeared in an earlier paper which de Moivre published in 1707.” ### 21.10 {#r-21.10} The fundamental theorem of algebra gives n complex roots counted with multiplicity for a degree-n polynomial; z²+1 has the two roots i and −i. d’Alembert attempted a proof in 1746; Gauss’s 1799 proof had a topological gap that Ostrowski filled only in 1920; Argand’s proof of 1806 was the first rigorous one and the first statement of the theorem for complex rather than real coefficients. Gauss’s 1799 proof was his doctoral dissertation (Helmstedt). *Status: confirmed* - [Fundamental theorem of algebra, Wikipedia](https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra) — states existence of a complex root and the degree-counting formulation with multiplicity, and its history section gives ‘A first attempt at proving the theorem was made by d’Alembert in 1746’, Gauss’s 1799 proof with ‘a topological gap, only filled by Alexander Ostrowski in 1920’, and ‘The first rigorous proof was published by Argand … in 1806 … it was also here that, for the first time, the fundamental theorem of algebra was stated for polynomials with complex coefficients’. - [NIST Digital Library of Mathematical Functions, §1.11](https://dlmf.nist.gov/1.11) — provides standard polynomial/root background; z²+1=(z−i)(z+i) is direct algebra. - [Carl Friedrich Gauss, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/) — the Duke of Brunswick “requested that Gauss submit a doctoral dissertation to the University of Helmstedt”; “Gauss’s dissertation was a discussion of the fundamental theorem of algebra.” ### 21.11 {#r-21.11} arg(i)=π/2 under the principal argument in (−π, π], and π/2 is transcendental because π is transcendental and nonzero rational scaling preserves transcendence. *Status: confirmed* - [Complex number, Wikipedia](https://en.wikipedia.org/wiki/Complex_number) — confirms the argument convention and that i lies at angle π/2. - [Lindemann–Weierstrass theorem, Wikipedia](https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem) — records the transcendence of π; if π/2 were algebraic, multiplying by 2 would make π algebraic. ### 21.12 {#r-21.12} The imaginary unit satisfies i²=−1; the historical card’s dates for Cardano, Descartes, Euler, Wessel, Argand, and Gauss describe the development of imaginary numbers and their geometric representation. Gauss in 1831 introduced the term complex number and suggested that direct, inverse and lateral units would have been better names than positive, negative and imaginary. *Status: judgement* - [Complex number, Wikipedia](https://en.wikipedia.org/wiki/Complex_number) — confirms the definition and gives the standard historical account of imaginary/complex numbers and their representations; “The term “imaginary” for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature.” Editorial note: The elementary definition is confirmed. The many exact historical attributions in the card are compressed chronology, and the fetched overview does not establish every individual date and priority claim at the wording’s precision; we leave them as a judgement call rather than silently upgrading them. ### 21.13 {#r-21.13} The extended complex plane adjoins one point at infinity, is conformally represented by the Riemann sphere, and Möbius transformations act on it; 1/0 is represented as infinity in the extended setting. The idea is Riemann’s, of 1857. *Status: confirmed* - [Riemann sphere, Wikipedia](https://en.wikipedia.org/wiki/Riemann_sphere) — confirms the one-point compactification, stereographic sphere model, Möbius transformations, and the extended-value convention 1/0 = ∞, and cites Riemann’s Theorie der Abel’schen Functionen of 1857 as the source of the construction. ### 21.14 {#r-21.14} The two-square identity is a norm-composition identity associated with Diophantus, Brahmagupta, and Fibonacci; it is the two-dimensional member of the composition-of-squares story. Brahmagupta’s identity is in the Brāhmasphuṭasiddhānta, written c. 628; Degen found the eight-square identity around 1818; Hurwitz proved in 1898 that such formulas exist only in 1, 2, 4 and 8 dimensions. *Status: judgement* - [Brahmagupta’s identity, Wikipedia](https://en.wikipedia.org/wiki/Brahmagupta%27s_identity) — confirms the identity and its connection with sums of two squares and Brahmagupta. - [Four-square identity, Wikipedia](https://en.wikipedia.org/wiki/Four-square_identity) — redirects to Euler’s four-square identity, which dates Euler’s letter to Goldbach 4 May 1748 and states that such an identity is possible only for n = 1, 2, 4 or 8 by Hurwitz’s theorem. - [Brahmagupta–Fibonacci identity, Wikipedia](https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity) — gives the identity, its first appearance in Diophantus’s Arithmetica (III, 19), Brahmagupta’s rediscovery and generalization to Pell’s equation, its introduction to western Europe by Fibonacci in the Book of Squares of 1225, the multiplicativity of the complex modulus it expresses, and the continuation through Euler’s four squares, Degen’s eight squares and Hurwitz’s classification, without their years; read again on 23 September 2026, the page carries neither 1818 nor 1898, which the Degen and Hurwitz lines carry; it dates Diophantus: “The identity first appeared in Diophantus’ Arithmetica (III, 19), of the third century A.D.” - [Brāhmasphuṭasiddhānta, Wikipedia](https://en.wikipedia.org/wiki/Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta) — read as raw wikitext: the work “is a main work of Brahmagupta, written c. 628,” and its contents include “Brahmagupta’s identity.” - [Degen’s eight-square identity, Wikipedia](https://en.wikipedia.org/wiki/Degen%27s_eight-square_identity) — read as raw wikitext: “First discovered by Carl Ferdinand Degen around 1818.” - [Hurwitz’s theorem (composition algebras), Wikipedia](https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)) — read as raw wikitext: “multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898.” Editorial note: The algebraic identity, its norm interpretation and the whole chain Diophantus → Brahmagupta → Fibonacci → Euler → Degen → Hurwitz are confirmed by the Brahmagupta–Fibonacci identity article. What that article does not carry is the year 628 for Brahmagupta, which the Brāhmasphuṭasiddhānta line now carries (the work was “written c. 628” and contains Brahmagupta’s identity). ### 21.15 {#r-21.15} The chapter’s mixed-type calculation is internally consistent: the numerator is three zeros, from ℂ, from the dual numbers and from a real-coordinate reading, each read as the scalar 0, and the denominator is the positive real 8+π/2, and the factor therefore returns the ordinary real integer 0. *Status: confirmed* - **Own computation** (the author's own work) — Using Euler’s identity, ε²=0, Re(i)=0, i·conjugate(i)=1, (1+i)(1−i)=2, |e^(iθ)|=1, j²=1, de Moivre=1, and #{z:z²+1=0}=2 gives numerator 0 and denominator 1+2+1+1+1+2+π/2=8+π/2>0; the quotient is 0. Editorial note: This explicitly checks the Part 2 mixed-type note: the algebra is performed in the stated systems, then the result comes home as an ordinary real number. ### 21.16 {#r-21.16} Bombelli’s L’algebra (1572) supplied multiplication rules for square roots of negative numbers and used them to solve particular cases of the irreducible cubic, obtaining its three real roots. Cardano called such a solution sophistic, ‘as subtle as it is useless’. *Status: confirmed* - [Veronica Gavagna, Rafael Bombelli, Treccani, Il Contributo italiano alla storia del Pensiero: Scienze](https://www.treccani.it/enciclopedia/rafael-bombelli_(Il-Contributo-italiano-alla-storia-del-Pensiero:-Scienze)/) — dates L’algebra to 1572 and explains that Cardano had encountered the irreducible cubic earlier, while Bombelli introduced new signs and multiplication rules for the negative-root quantities and thereby obtained the three real roots in particular cases. - [Ars Magna (Cardano book), Wikipedia](https://en.wikipedia.org/wiki/Ars_Magna_(Cardano_book)) — read as raw wikitext: Cardano called the solution “sophistic,” and “Cardano then says that this answer is “as subtle as it is useless,”” the answer being 5 ± √−15; and “The Ars Magna (The Great Art, 1545) is an important Latin-language book on algebra written by Gerolamo Cardano.” Editorial note: Added in the V14 prose pass. The prose had called Bombelli the first person to need the trick, while the chapter’s own imaginary-unit card says Cardano had already computed with such roots in 1545. Treccani distinguishes the two episodes: Cardano encountered the obstacle; Bombelli made an arithmetic of the quantities and solved particular irreducible cubics. The prose now says Bombelli made the trick work, not that he encountered it first. ### 21.17 {#r-21.17} The chapter epigraph is Benjamin Peirce’s remark to his Harvard class on Euler’s identity, from notes by his student W. E. Byerly published in R. C. Archibald’s Benjamin Peirce, 1809-1880: Biographical Sketch and Bibliography of 1925, and also in Kasner and Newman’s Mathematics and the Imagination of 1940. *Status: confirmed* - [Benjamin Peirce, Wikiquote](https://en.wikiquote.org/wiki/Benjamin_Peirce) — read as raw HTML: “Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth. On Euler’s identity, e^(iπ) + 1 = 0. as quoted in notes by W. E. Byerly, published in Benjamin Peirce, 1809-1880: Biographical Sketch and Bibliography (1925) by R. C. Archibald; also in Mathematics and the Imagination (1940) by Edward Kasner and James Newman,” which is the footer line word for word. Editorial note: Added 17 September 2026 to source the chapter epigraph, which carried no marker. The last of the four secondhand ones: what survives is a student’s notes printed forty-five years later, and the claim line says so. ### 21.18 {#r-21.18} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Lindemann proved pi transcendental in 1882; Descartes coined the word ‘imaginary’ in La Géométrie (1637); Newton was using the word ‘imaginary’ by 1670; Euler introduced the letter i in a paper of 1777, printed 1794. *Status: confirmed* - [Lindemann–Weierstrass theorem, Wikipedia](https://en.wikipedia.org/w/index.php?title=Lindemann%E2%80%93Weierstrass_theorem) — read raw: “Lindemann proved in 1882 that e^α is transcendental for every non-zero algebraic number α, thereby establishing that π is transcendental (see below)..” - [Earliest Known Uses of Some of the Words of Mathematics (I), Jeff Miller, hosted by MacTutor](https://mathshistory.st-andrews.ac.uk/Miller/mathword/i/) — read raw: “The terms IMAGINARY and REAL were introduced in French by Rene Descartes (1596-1650) in “La Geometrie” (1637):.” - [Imaginary unit, Wikipedia](https://en.wikipedia.org/w/index.php?title=Imaginary_unit) — read raw: “The name imaginary is generally credited to René Descartes, and Isaac Newton used the term as early as 1670..” - [Earliest Uses of Symbols for Constants, Jeff Miller, hosted by MacTutor](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/constants/) — read raw: “This was first used by Leonhard Euler (1707-1783) in a memoir presented in 1777 but not published until 1794 in his “Institutionum calculi integralis..” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 21.20 {#r-21.20} States Fermat’s theorem on sums of two squares, that an odd prime is a sum of two squares exactly when it is congruent to 1 mod 4, stated by Fermat around 1640 and first proved by Euler, who announced his proof in 1747 and 1749 and published it between 1752 and 1755. Its smallest examples are 5 and 13 = 2² + 3², and the representation is unique up to order and sign. *Status: confirmed* - [Fermat’s theorem on sums of two squares, Wikipedia](https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares) — states the theorem and its p ≡ 1 (mod 4) condition, dates Fermat’s claim to 1640, and says Euler announced a proof in letters to Goldbach of 1747 and 1749, succeeded in 1749 and published it in 1752–55; “Fermat’s theorem on sums of two squares is strongly related with the theory of Gaussian primes,” and “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 p = 4k + 3, and that the second case occurs when p = 4k + 1 and p = 2.” - [Pick’s theorem and sums of two squares, Keith Conrad (UConn)](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/Picksumofsq.pdf) — states uniqueness: “Furthermore, a representation of a prime as a2 +b2 in Z is unique up to the order and signs of a and b.”