Prime number (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Prime number" in English language version.

Last modified: 12:44:57, 12 August 2026 (UTC)

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Pomerance, Carl (December 1982). "The Search for Prime Numbers". Scientific American. 247 (6): 136–147. Bibcode:1982SciAm.247f.136P. doi:10.1038/scientificamerican1282-136. JSTOR 24966751.
Atkin, A O.L.; Morain, F. (1993). "Elliptic curves and primality proving" (PDF). Mathematics of Computation. 61 (203): 29–68. Bibcode:1993MaCom..61...29A. doi:10.1090/s0025-5718-1993-1199989-x. JSTOR 2152935. MR 1199989.
Morain, F. (2007). "Implementing the asymptotically fast version of the elliptic curve primality proving algorithm". Mathematics of Computation. 76 (257): 493–505. arXiv:math/0502097. Bibcode:2007MaCom..76..493M. doi:10.1090/S0025-5718-06-01890-4. MR 2261033. S2CID 133193.
Martín-López, Enrique; Laing, Anthony; Lawson, Thomas; Alvarez, Roberto; Zhou, Xiao-Qi; O'Brien, Jeremy L. (12 October 2012). "Experimental realization of Shor's quantum factoring algorithm using qubit recycling". Nature Photonics. 6 (11): 773–776. arXiv:1111.4147. Bibcode:2012NaPho...6..773M. doi:10.1038/nphoton.2012.259. S2CID 46546101.
Zhu, Huangjun (2010). "SIC POVMs and Clifford groups in prime dimensions". Journal of Physics A: Mathematical and Theoretical. 43 (30) 305305. arXiv:1003.3591. Bibcode:2010JPhA...43D5305Z. doi:10.1088/1751-8113/43/30/305305. S2CID 118363843.
Goles, E.; Schulz, O.; Markus, M. (2001). "Prime number selection of cycles in a predator-prey model". Complexity. 6 (4): 33–38. Bibcode:2001Cmplx...6d..33G. doi:10.1002/cplx.1040.
Campos, Paulo R. A.; de Oliveira, Viviane M.; Giro, Ronaldo; Galvão, Douglas S. (2004). "Emergence of prime numbers as the result of evolutionary strategy". Physical Review Letters. 93 (9) 098107. arXiv:q-bio/0406017. Bibcode:2004PhRvL..93i8107C. doi:10.1103/PhysRevLett.93.098107. PMID 15447148. S2CID 88332.

ams.org (Global: 434^th place; English: 249^th place)

mathscinet.ams.org

Ziegler, Günter M. (2004). "The great prime number record races". Notices of the American Mathematical Society. 51 (4): 414–416. MR 2039814.
Sierpiński, Wacław (1964). A Selection of Problems in the Theory of Numbers. New York: Macmillan. p. 40. MR 0170843.
Nathanson, Melvyn B. (2000). "Notations and Conventions". Elementary Methods in Number Theory. Graduate Texts in Mathematics. Vol. 195. Springer. ISBN 978-0-387-22738-2. MR 1732941.
Knorr, Wilbur (1982). "Techniques of fractions in ancient Egypt and Greece". Historia Mathematica. 9 (2): 133–171. doi:10.1016/0315-0860(82)90001-5. MR 0662138. See p. 136, where Knorr writes (of the Rhind papyrus) "There are two methods, depending on whether $n$ has proper divisors or not."
Mollin, Richard A. (2002). "A brief history of factoring and primality testing B. C. (before computers)". Mathematics Magazine. 75 (1): 18–29. doi:10.2307/3219180. JSTOR 3219180. MR 2107288.
Apostol, Tom M. (2000). "A centennial history of the prime number theorem". In Bambah, R.P.; Dumir, V.C.; Hans-Gill, R.J. (eds.). Number Theory. Trends in Mathematics. Basel: Birkhäuser. pp. 1–14. MR 1764793.
Apostol, Tom M. (1976). "7. Dirichlet's Theorem on Primes in Arithmetical Progressions". Introduction to Analytic Number Theory. New York; Heidelberg: Springer-Verlag. pp. 146–156. MR 0434929.
Katz, Shaul (2004). "Berlin roots – Zionist incarnation: the ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem". Science in Context. 17 (1–2): 199–234. doi:10.1017/S0269889704000092. MR 2089305. S2CID 145575536.
Caldwell, Chris K.; Reddick, Angela; Xiong, Yeng; Keller, Wilfrid (2012). "The history of the primality of one: a selection of sources". Journal of Integer Sequences. 15 (9): Article 12.9.8. MR 3005523. Archived from the original on 2018-04-12. Retrieved 2018-01-15. For a selection of quotes from and about the ancient Greek positions on the status of 1 and 2, see in particular pp. 3–4. For the Islamic mathematicians, see p. 6.
Caldwell, Chris K.; Xiong, Yeng (2012). "What is the smallest prime?" (PDF). Journal of Integer Sequences. 15 (9): Article 12.9.7. MR 3005530. Archived (PDF) from the original on 2018-04-12. Retrieved 2018-01-15.
Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization (2nd ed.). Basel, Switzerland: Birkhäuser. p. 36. doi:10.1007/978-1-4612-0251-6. ISBN 978-0-8176-3743-9. MR 1292250.
Conway, John Horton; Guy, Richard K. (1996). The Book of Numbers. New York: Copernicus. pp. 129–130. doi:10.1007/978-1-4612-4072-3. ISBN 978-0-387-97993-9. MR 1411676.
Furstenberg, Harry (1955). "On the infinitude of primes". American Mathematical Monthly. 62 (5): 353. doi:10.2307/2307043. JSTOR 2307043. MR 0068566.
Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio (2014). "Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\cdot 10^{18}$ ". Mathematics of Computation. 83 (288): 2033–2060. doi:10.1090/S0025-5718-2013-02787-1. MR 3194140.
Ramaré, Olivier (1995). "On Šnirel'man's constant". Annali della Scuola Normale Superiore di Pisa. 22 (4): 645–706. MR 1375315. Archived from the original on 2022-02-09. Retrieved 2018-01-23.
Rassias, Michael Th. (2017). Goldbach's Problem: Selected Topics. Cham: Springer. p. vii. doi:10.1007/978-3-319-57914-6. ISBN 978-3-319-57912-2. MR 3674356.
Hua, L. K. (2009) [1965]. Additive Theory of Prime Numbers. Translations of Mathematical Monographs. Vol. 13. Providence, RI: American Mathematical Society. pp. 176–177. ISBN 978-0-8218-4942-2. MR 0194404. OCLC 824812353.
Patterson, S. J. (1988). An introduction to the theory of the Riemann zeta-function. Cambridge Studies in Advanced Mathematics. Vol. 14. Cambridge University Press, Cambridge. p. 1. doi:10.1017/CBO9780511623707. ISBN 978-0-521-33535-5. MR 0933558.
Borwein, Peter; Choi, Stephen; Rooney, Brendan; Weirathmueller, Andrea (2008). The Riemann hypothesis: A resource for the afficionado and virtuoso alike. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. New York: Springer. pp. 10–11. doi:10.1007/978-0-387-72126-2. ISBN 978-0-387-72125-5. MR 2463715.
Childress, Nancy (2009). Class Field Theory. Universitext. Springer, New York. pp. 8–11. doi:10.1007/978-0-387-72490-4. ISBN 978-0-387-72489-8. MR 2462595. See also p. 64.
Erickson, Marty; Vazzana, Anthony; Garth, David (2016). Introduction to Number Theory. Textbooks in Mathematics (2nd ed.). Boca Raton, FL: CRC Press. p. 200. ISBN 978-1-4987-1749-6. MR 3468748.
Weil, André (1995). Basic Number Theory. Classics in Mathematics. Berlin: Springer-Verlag. p. 43. ISBN 978-3-540-58655-5. MR 1344916. Note however that some authors such as Childress (2009) instead use "place" to mean an equivalence class of norms.
Koch, H. (1997). Algebraic Number Theory. Berlin: Springer-Verlag. p. 136. doi:10.1007/978-3-642-58095-6. ISBN 978-3-540-63003-6. MR 1474965.
Lauritzen, Niels (2003). Concrete Abstract Algebra: From numbers to Gröbner bases. Cambridge: Cambridge University Press. p. 127. doi:10.1017/CBO9780511804229. ISBN 978-0-521-53410-9. MR 2014325.
Eisenbud, David (1995). Commutative Algebra. Graduate Texts in Mathematics. Vol. 150. Berlin; New York: Springer-Verlag. Section 3.3. doi:10.1007/978-1-4612-5350-1. ISBN 978-0-387-94268-1. MR 1322960.
Shafarevich, Igor R. (2013). "Definition of $\operatorname {Spec} A$ ". Basic Algebraic Geometry 2: Schemes and Complex Manifolds (3rd ed.). Springer, Heidelberg. p. 5. doi:10.1007/978-3-642-38010-5. ISBN 978-3-642-38009-9. MR 3100288.
Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 322. Berlin: Springer-Verlag. Section I.8, p. 50. doi:10.1007/978-3-662-03983-0. ISBN 978-3-540-65399-8. MR 1697859.
Stevenhagen, P.; Lenstra, H.W. Jr. (1996). "Chebotarëv and his density theorem". The Mathematical Intelligencer. 18 (2): 26–37. doi:10.1007/BF03027290. MR 1395088. S2CID 14089091.
Hromkovič, Juraj (2001). "5.5 Bibliographic Remarks". Algorithmics for Hard Problems. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin. pp. 383–385. doi:10.1007/978-3-662-04616-6. ISBN 978-3-540-66860-2. MR 1843669. S2CID 31159492.
Koblitz, Neal (1987). "Chapter V. Primality and Factoring". A Course in Number Theory and Cryptography. Graduate Texts in Mathematics. Vol. 114. Springer-Verlag, New York. pp. 112–149. doi:10.1007/978-1-4684-0310-7_5. ISBN 978-0-387-96576-5. MR 0910297.
Tao, Terence (2010). "1.11 The AKS primality test". An epsilon of room, II: Pages from year three of a mathematical blog. Graduate Studies in Mathematics. Vol. 117. Providence, RI: American Mathematical Society. pp. 82–86. doi:10.1090/gsm/117. ISBN 978-0-8218-5280-4. MR 2780010. Archived from the original on 2018-01-19. Retrieved 2018-01-18.
Atkin, A O.L.; Morain, F. (1993). "Elliptic curves and primality proving" (PDF). Mathematics of Computation. 61 (203): 29–68. Bibcode:1993MaCom..61...29A. doi:10.1090/s0025-5718-1993-1199989-x. JSTOR 2152935. MR 1199989.
Morain, F. (2007). "Implementing the asymptotically fast version of the elliptic curve primality proving algorithm". Mathematics of Computation. 76 (257): 493–505. arXiv:math/0502097. Bibcode:2007MaCom..76..493M. doi:10.1090/S0025-5718-06-01890-4. MR 2261033. S2CID 133193.
Lenstra, H. W. Jr.; Pomerance, Carl (2019). "Primality testing with Gaussian periods" (PDF). Journal of the European Mathematical Society. 21 (4): 1229–1269. doi:10.4171/JEMS/861. hdl:21.11116/0000-0005-717D-0. MR 3941463. S2CID 127807021. Archived (PDF) from the original on 2023-12-27. Retrieved 2018-01-18.
Baillie, Robert; Wagstaff, Jr., Samuel S. (October 1980). "Lucas Pseudoprimes" (PDF). Mathematics of Computation. 35 (152): 1391–1417. doi:10.1090/S0025-5718-1980-0583518-6. JSTOR 2006406. MR 0583518. Archived (PDF) from the original on 2016-03-04. Retrieved 2019-05-29.
Monier, Louis (1980). "Evaluation and comparison of two efficient probabilistic primality testing algorithms". Theoretical Computer Science. 12 (1): 97–108. doi:10.1016/0304-3975(80)90007-9. MR 0582244.
Tao, Terence (2009). "1.7 The Lucas–Lehmer test for Mersenne primes". Poincaré's legacies, pages from year two of a mathematical blog. Part I. Providence, RI: American Mathematical Society. pp. 36–41. ISBN 978-0-8218-4883-8. MR 2523047. Archived from the original on 2017-08-07. Retrieved 2018-01-19.
Pomerance, Carl (1996). "A tale of two sieves". Notices of the American Mathematical Society. 43 (12): 1473–1485. MR 1416721.
Roth, Klaus F. (1951). "On a problem of Heilbronn". Journal of the London Mathematical Society. Second Series. 26 (3): 198–204. doi:10.1112/jlms/s1-26.3.198. MR 0041889.
Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211. Berlin, Germany; New York: Springer-Verlag. doi:10.1007/978-1-4613-0041-0. ISBN 978-0-387-95385-4. MR 1878556. Section II.1, p. 90.
Schubert, Horst (1949). "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten". S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (3): 57–104. MR 0031733.
Milnor, J. (1962). "A unique decomposition theorem for 3-manifolds". American Journal of Mathematics. 84 (1): 1–7. doi:10.2307/2372800. JSTOR 2372800. MR 0142125.
Křížek, Michal; Luca, Florian; Somer, Lawrence (2001). 17 Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics. Vol. 9. New York: Springer-Verlag. pp. 1–2. doi:10.1007/978-0-387-21850-2. ISBN 978-0-387-95332-8. MR 1866957.
Gleason, Andrew M. (1988). "Angle trisection, the heptagon, and the triskaidecagon". American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624. JSTOR 2323624. MR 0935432.
Ziegler, Günter M. (2015). "Cannons at sparrows". European Mathematical Society Newsletter (95): 25–31. MR 3330472.
Pomerance, Carl (2004). "Prime Numbers and the Search for Extraterrestrial Intelligence" (PDF). In Hayes, David F.; Ross, Peter (eds.). Mathematical Adventures for Students and Amateurs. MAA Spectrum. Washington, DC: Mathematical Association of America. pp. 3–6. ISBN 978-0-88385-548-5. MR 2085842. Archived (PDF) from the original on 2019-03-23. Retrieved 2018-01-27.

ams.org

Atkin, A O.L.; Morain, F. (1993). "Elliptic curves and primality proving" (PDF). Mathematics of Computation. 61 (203): 29–68. Bibcode:1993MaCom..61...29A. doi:10.1090/s0025-5718-1993-1199989-x. JSTOR 2152935. MR 1199989.

archive.org (Global: 6^th place; English: 6^th place)

Gardiner, Anthony (1997). The Mathematical Olympiad Handbook: An Introduction to Problem Solving Based on the First 32 British Mathematical Olympiads 1965–1996. Oxford University Press. p. 26. ISBN 978-0-19-850105-3.
Adler, Irving (1960). The Giant Golden Book of Mathematics: Exploring the World of Numbers and Space. Golden Press. p. 16. OCLC 6975809.
Leff, Lawrence S. (2000). Math Workbook for the SAT I. Barron's Educational Series. p. 360. ISBN 978-0-7641-0768-9.
Dudley, Underwood (1978). "Section 2: Unique factorization". Elementary number theory (2nd ed.). W.H. Freeman and Co. p. 10. ISBN 978-0-7167-0076-0.
Sierpiński, Wacław (1964). A Selection of Problems in the Theory of Numbers. New York: Macmillan. p. 40. MR 0170843.
Conway, John Horton; Guy, Richard K. (1996). The Book of Numbers. New York: Copernicus. pp. 129–130. doi:10.1007/978-1-4612-4072-3. ISBN 978-0-387-97993-9. MR 1411676.
Leff 2000, p. 64–65.
du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. p. 23. ISBN 978-0-06-093558-0.
Furstenberg, Harry (1955). "On the infinitude of primes". American Mathematical Monthly. 62 (5): 353. doi:10.2307/2307043. JSTOR 2307043. MR 0068566.
Wright, E.M. (1951). "A prime-representing function". American Mathematical Monthly. 58 (9): 616–618. doi:10.2307/2306356. JSTOR 2306356.
Weil, André (1995). Basic Number Theory. Classics in Mathematics. Berlin: Springer-Verlag. p. 43. ISBN 978-3-540-58655-5. MR 1344916. Note however that some authors such as Childress (2009) instead use "place" to mean an equivalence class of norms.
Giblin, Peter (1993). Primes and Programming. Cambridge University Press. p. 39. ISBN 978-0-521-40988-9.
Giblin 1993, p. 54

archives-ouvertes.fr (Global: 1,143^rd place; English: 973^rd place)

hal-univ-paris8.archives-ouvertes.fr

Bullynck, Maarten (2010). "A history of factor tables with notes on the birth of number theory 1657–1817". Revue d'Histoire des Mathématiques. 16 (2): 133–216. Archived from the original on 2023-06-04. Retrieved 2018-01-17.

arxiv.org (Global: 49^th place; English: 31^st place)

Green, Ben; Tao, Terence (2008). "The primes contain arbitrarily long arithmetic progressions". Annals of Mathematics. 167 (2): 481–547. arXiv:math.NT/0404188. doi:10.4007/annals.2008.167.481. S2CID 1883951.
Farach-Colton, Martín; Tsai, Meng-Tsung (2015). "On the complexity of computing prime tables". In Elbassioni, Khaled; Makino, Kazuhisa (eds.). Algorithms and Computation: 26th International Symposium, ISAAC 2015, Nagoya, Japan, December 9-11, 2015, Proceedings. Lecture Notes in Computer Science. Vol. 9472. Springer. pp. 677–688. arXiv:1504.05240. doi:10.1007/978-3-662-48971-0_57. ISBN 978-3-662-48970-3.
Morain, F. (2007). "Implementing the asymptotically fast version of the elliptic curve primality proving algorithm". Mathematics of Computation. 76 (257): 493–505. arXiv:math/0502097. Bibcode:2007MaCom..76..493M. doi:10.1090/S0025-5718-06-01890-4. MR 2261033. S2CID 133193.
Martín-López, Enrique; Laing, Anthony; Lawson, Thomas; Alvarez, Roberto; Zhou, Xiao-Qi; O'Brien, Jeremy L. (12 October 2012). "Experimental realization of Shor's quantum factoring algorithm using qubit recycling". Nature Photonics. 6 (11): 773–776. arXiv:1111.4147. Bibcode:2012NaPho...6..773M. doi:10.1038/nphoton.2012.259. S2CID 46546101.
Boklan, Kent D.; Conway, John H. (January 2017). "Expect at most one billionth of a new Fermat prime!". The Mathematical Intelligencer. 39 (1): 3–5. arXiv:1605.01371. doi:10.1007/s00283-016-9644-3. S2CID 119165671.
Zhu, Huangjun (2010). "SIC POVMs and Clifford groups in prime dimensions". Journal of Physics A: Mathematical and Theoretical. 43 (30) 305305. arXiv:1003.3591. Bibcode:2010JPhA...43D5305Z. doi:10.1088/1751-8113/43/30/305305. S2CID 118363843.
Campos, Paulo R. A.; de Oliveira, Viviane M.; Giro, Ronaldo; Galvão, Douglas S. (2004). "Emergence of prime numbers as the result of evolutionary strategy". Physical Review Letters. 93 (9) 098107. arXiv:q-bio/0406017. Bibcode:2004PhRvL..93i8107C. doi:10.1103/PhysRevLett.93.098107. PMID 15447148. S2CID 88332.

biodiversitylibrary.org (Global: 168^th place; English: 136^th place)

Kummer, Ernst (1878-11-25). "Neuer elementarer Beweis des Satzes, dass die Anzahl aller Primzahlen eine unendliche ist". Monatsberichte der Königlichen Preussische Akademie des Wissenschaften zu Berlin (in German): 777–778.

books.google.com (Global: 3^rd place; English: 3^rd place)

Henderson, Anne (2014). Dyslexia, Dyscalculia and Mathematics: A practical guide (2nd ed.). Routledge. p. 62. ISBN 978-1-136-63662-2.
Dudley, Underwood (1978). "Section 2: Unique factorization". Elementary number theory (2nd ed.). W.H. Freeman and Co. p. 10. ISBN 978-0-7167-0076-0.
Sierpiński, Wacław (1988). Elementary Theory of Numbers. North-Holland Mathematical Library. Vol. 31 (2nd ed.). Elsevier. p. 113. ISBN 978-0-08-096019-7.
Stillwell, John (1997). Numbers and Geometry. Undergraduate Texts in Mathematics. Springer. p. 9. ISBN 978-0-387-98289-2.
Nathanson, Melvyn B. (2000). "Notations and Conventions". Elementary Methods in Number Theory. Graduate Texts in Mathematics. Vol. 195. Springer. ISBN 978-0-387-22738-2. MR 1732941.
Faticoni, Theodore G. (2012). The Mathematics of Infinity: A Guide to Great Ideas. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Vol. 111 (2nd ed.). John Wiley & Sons. p. 44. ISBN 978-1-118-24382-4.
Stillwell, John (2010). Mathematics and Its History. Undergraduate Texts in Mathematics (3rd ed.). Springer. p. 40. ISBN 978-1-4419-6052-8.
Sandifer 2007, 8. Fermat's Little Theorem (November 2003), p. 45
Sandifer, C. Edward (2014). How Euler Did Even More. Mathematical Association of America. p. 42. ISBN 978-0-88385-584-3.
Koshy, Thomas (2002). Elementary Number Theory with Applications. Academic Press. p. 369. ISBN 978-0-12-421171-1.
Yuan, Wang (2002). Goldbach Conjecture. Series In Pure Mathematics. Vol. 4 (2nd ed.). World Scientific. p. 21. ISBN 978-981-4487-52-8.
Narkiewicz, Wladyslaw (2000). "1.2 Sum of Reciprocals of Primes". The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Springer Monographs in Mathematics. Springer. p. 11. ISBN 978-3-540-66289-1.
Apostol, Tom M. (2000). "A centennial history of the prime number theorem". In Bambah, R.P.; Dumir, V.C.; Hans-Gill, R.J. (eds.). Number Theory. Trends in Mathematics. Basel: Birkhäuser. pp. 1–14. MR 1764793.
Apostol, Tom M. (1976). "7. Dirichlet's Theorem on Primes in Arithmetical Progressions". Introduction to Analytic Number Theory. New York; Heidelberg: Springer-Verlag. pp. 146–156. MR 0434929.
Chabert, Jean-Luc (2012). A History of Algorithms: From the Pebble to the Microchip. Springer. p. 261. ISBN 978-3-642-18192-4.
Cooper, S. Barry; Hodges, Andrew (2016). The Once and Future Turing. Cambridge University Press. pp. 37–38. ISBN 978-1-107-01083-3.
Beiler, Albert H. (1999) [1966]. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover. p. 2. ISBN 978-0-486-21096-4. OCLC 444171535.
Kraft, James S.; Washington, Lawrence C. (2014). Elementary Number Theory. Textbooks in mathematics. CRC Press. p. 7. ISBN 978-1-4987-0269-0.
Bauer, Craig P. (2013). Secret History: The Story of Cryptology. Discrete Mathematics and Its Applications. CRC Press. p. 468. ISBN 978-1-4665-6186-1.
Klee, Victor; Wagon, Stan (1991). Old and New Unsolved Problems in Plane Geometry and Number Theory. Dolciani mathematical expositions. Vol. 11. Cambridge University Press. p. 224. ISBN 978-0-88385-315-3.
Tarán, Leonardo (1981). Speusippus of Athens: A Critical Study With a Collection of the Related Texts and Commentary. Philosophia Antiqua : A Series of Monographs on Ancient Philosophy. Vol. 39. Brill. pp. 35–38. ISBN 978-90-04-06505-5.
Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization (2nd ed.). Basel, Switzerland: Birkhäuser. p. 36. doi:10.1007/978-1-4612-0251-6. ISBN 978-0-8176-3743-9. MR 1292250.
For the totient, see Sierpiński 1988, p. 245. For the sum of divisors, see Sandifer, C. Edward (2007). How Euler Did It. MAA Spectrum. Mathematical Association of America. p. 59. ISBN 978-0-88385-563-8.
Smith, Karl J. (2011). The Nature of Mathematics (12th ed.). Cengage Learning. p. 188. ISBN 978-0-538-73758-6.
Dudley 1978, Section 2, Theorem 2, p. 16; Neale, Vicky (2017). Closing the Gap: The Quest to Understand Prime Numbers. Oxford University Press. p. 107. ISBN 978-0-19-109243-5.
Dudley 1978, Section 2, Lemma 5, p. 15; Higgins, Peter M. (1998). Mathematics for the Curious. Oxford University Press. pp. 77–78. ISBN 978-0-19-150050-3.
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archives-ouvertes.fr (Global: 1,143^rd place; English: 973^rd place)

arxiv.org (Global: 49^th place; English: 31^st place)

biodiversitylibrary.org (Global: 168^th place; English: 136^th place)

books.google.com (Global: 3^rd place; English: 3^rd place)

clarku.edu (Global: 9,653^rd place; English: 8,559^th place)

dartmouth.edu (Global: 2,786^th place; English: 1,776^th place)

doi.org (Global: 2^nd place; English: 2^nd place)

economist.com (Global: 331^st place; English: 272^nd place)

eff.org (Global: 4,491^st place; English: 3,102^nd place)

elsevier.com (Global: 365^th place; English: 372^nd place)

free.fr (Global: 283^rd place; English: 705^th place)

hathitrust.org (Global: 296^th place; English: 153^rd place)

hdl.handle.net (Global: 66^th place; English: 34^th place)

ietf.org (Global: 731^st place; English: 625^th place)

iop.org (Global: 1,046^th place; English: 1,608^th place)

jstor.org (Global: 23^rd place; English: 15^th place)

maa.org (Global: 4,358^th place; English: 2,907^th place)

nationalgeographic.com (Global: 489^th place; English: 425^th place)

nih.gov (Global: 5^th place; English: 5^th place)

nodak.edu (Global: 4,368^th place; English: 5,513^th place)

numdam.org (Global: 8,359^th place; English: 6,894^th place)

nytimes.com (Global: 7^th place; English: 7^th place)

oeis.org (Global: 1,188^th place; English: 866^th place)

rfc-editor.org (Global: 747^th place; English: 325^th place)

sciencenews.org (Global: 4,656^th place; English: 3,908^th place)

semanticscholar.org (Global: 15^th place; English: 8^th place)

st-andrews.ac.uk (Global: 1,053^rd place; English: 1,274^th place)

theguardian.com (Global: 16^th place; English: 11^th place)

theregister.co.uk (Global: 1,877^th place; English: 1,233^rd place)

usc.edu (Global: 1,432^nd place; English: 908^th place)

utm.edu (Global: 1,830^th place; English: 1,501^st place)

uwaterloo.ca (Global: 3,649^th place; English: 2,434^th place)

web.archive.org (Global: 1^st place; English: 1^st place)

worldcat.org (Global: 4^th place; English: 4^th place)