수론

3. 분야

현대 수론은 크게 다음과 같이 분류할 수 있다.

• '''초등 정수론''': 복소해석학을 사용하지 않고, 페르마 소정리, 오일러의 정리 등과 같이 수론의 가장 기초적인 토대를 이루는 분야이다.
• '''해석적 수론''': 복소해석학 등의 해석학적 기법을 사용하여 정수의 분포, 밀도 등을 연구한다. 소수 정리, 리만 가설 등이 주요 연구 대상이다.
• '''대수적 수론''': 대수적 수체를 연구하며, 추상대수학의 기법을 사용하여 아이디얼, 유수 등 대수적 구조를 통해 정수의 성질을 연구한다.
• '''계산 수론''': 소인수 분해나 유한체에 대한 알고리즘 연구와 같이 수론에서 등장하는 값들을 계산하는 알고리즘을 연구한다.
• '''디오판토스 기하학''': 디오판토스 방정식을 대수기하학적 관점에서 연구하며, 타원 곡선, 아벨 다양체 등 대수다양체의 정수점, 유리점 등을 연구한다.

3. 1. 초등 정수론

3. 2. 해석적 수론

3. 3. 대수적 수론

3. 4. 계산 수론

3. 5. 디오판토스 기하학

3. 6. 기타 분야

4. 응용

수론은 암호학, 컴퓨터 과학, 물리학 등 다양한 분야에 응용된다. 과거 수론학자 레너드 딕슨은 수론이 응용 분야에 물들지 않았다고 언급했지만,^[52] 현대에는 이러한 관점이 더 이상 유효하지 않다. 1974년 도널드 커누스는 "기본 수론의 거의 모든 정리는 컴퓨터가 고속 수치 계산을 수행하는 문제와 관련하여 자연스럽고 동기 부여된 방식으로 발생한다"라고 말했다.^[53]

수론은 RSA와 같은 공개 키 암호화 방식, 고속 푸리에 변환 알고리즘, 오류 정정 코드, 통신, 음악의 음계 등 다양한 분야에 활용된다. 물리학에서 리만 가설은 소수의 분포와 관련하여 연구되고 있다.^[57]

4. 1. 암호학

4. 2. 컴퓨터 과학

4. 3. 기타 응용

5. 미해결 문제

수론에는 아직 해결되지 않은 많은 문제들이 남아 있다. 리만 가설, 골드바흐의 추측, 쌍둥이 소수 추측, BSD 추측 등이 대표적인 예이다.^[50] 이러한 문제들은 소수의 분포와 관련된 경우가 많으며, 문제 자체는 쉽게 이해할 수 있지만, 그 증명은 매우 어려운 경우가 많다.

참조

_[1] 서적 "By arithmetic, Plato meant, not arithmetic in our sense, but the science which considers numbers in themselves, in other words, what we mean by the Theory of Numbers."
_[2] 서적 Take, for example, In 1952, still had to specify that he meant ''The Higher Arithmetic''. Hardy and Wright wrote in the introduction to ''An Introduction to the Theory of Numbers'' (1938): "We proposed at one time to change [the title] to ''An introduction to arithmetic'', a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book."
_[3] 서적 The term ''takiltum'' is problematic. Robson prefers the rendering "The holding-square of the diagonal from which 1 is torn out, so that the short side comes up...".
_[4] 서적 Other sources give the modern formula $(p^2-q^2,2pq,p^2+q^2)$. Van der Waerden gives both the modern formula and what amounts to the form preferred by Robson.
_[5] 서적 Neugebauer discusses the table in detail and mentions in passing Euclid's method in modern notation.
_[6] 서적 This is controversial. See [[Plimpton 322]]. Robson's article is written polemically with a view to "perhaps [...] knocking [Plimpton 322] off its pedestal"; at the same time, it settles to the conclusion that

[...] the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems.

_[14] 서적
_[15] 서적
_[16] 서적 See, for example, ''Sunzi Suanjing'', Ch. 3, Problem 36, in :

_[17] 웹사이트 Eusebius of Caesarea: Praeparatio Evangelica (Preparation for the Gospel). Tr. E.H. Gifford (1903) – Book 10 http://www.tertullia[...] 2017-02-20
_[18] 문서 Metaphysics, 1.6.1 (987a)
_[19] 문서 Tusc. Disput. 1.17.39.
_[20] 서적 Any early contact between Babylonian and Indian mathematics remains conjectural.
_[21] 서적 Āryabhaṭa, Āryabhatīya, Chapter 2, verses 32–33, cited in: See also A slightly more explicit description of the kuṭṭaka was later given in [[Brahmagupta]], ''Brāhmasphuṭasiddhānta'', XVIII, 3–5 (in , cited in ).
_[22] 서적 , cited in See also the preface in cited in
_[23] 서적 and , cited in .
_[24] 문서 [[Claude Gaspard Bachet de Méziriac|Bachet]], 1621, following a first attempt by [[Guilielmus Xylander|Xylander]], 1575
_[25] 서적 This was more so in number theory than in other areas (remark in ). Bachet's own proofs were "ludicrously clumsy".
_[26] 서적 Perfect and especially amicable numbers are of little or no interest nowadays. The same was not true in medieval times—whether in the West or the Arab-speaking world—due in part to the importance given to them by the Neopythagorean (and hence mystical) [[Nicomachus of Gerasa|Nicomachus]] (ca. 100 CE), who wrote a primitive but influential "[[Introduction to Arithmetic]]". See .
_[27] 서적 The initial subjects of Fermat's correspondence included divisors ("aliquot parts") and many subjects outside number theory; see the list in the letter from Fermat to Roberval, 22.IX.1636, , cited in .
_[28] 서적 Numbers and Measurements https://books.google[...] Encyclopaedia Britannica 2017
_[29] 서적 , Letter XLVI from Fermat to Frenicle, 1640, cited in
_[30] 서적 Here, as usual, given two integers ''a'' and ''b'' and a non-zero integer ''m'', we write $a\; \backslash equiv\; b\; \backslash bmod\; m$ (read "''a'' is congruent to ''b'' modulo ''m''") to mean that ''m'' divides ''a'' − ''b'', or, what is the same, ''a'' and ''b'' leave the same residue when divided by ''m''. This notation is actually much later than Fermat's; it first appears in section 1 of [[Gauss]]'s [[Disquisitiones Arithmeticae]]. Fermat's little theorem is a consequence of the [[Lagrange's theorem (group theory)|fact]] that the [[Order (group theory)|order]] of an element of a group divides the [[Order (group theory)|order]] of the group. The modern proof would have been within Fermat's means (and was indeed given later by Euler), even though the modern concept of a group came long after Fermat or Euler. (It helps to know that inverses exist modulo ''p'', that is, given ''a'' not divisible by a prime ''p'', there is an integer ''x'' such that $x\; a\; \backslash equiv\; 1\; \backslash bmod\; p$); this fact (which, in modern language, makes the residues mod ''p'' into a group, and which was already known to Āryabhaṭa; see [[#Indian school: Āryabhaṭa, Brahmagupta, Bhāskara|above]]) was familiar to Fermat thanks to its rediscovery by [[Claude Gaspard Bachet de Méziriac|Bachet]] Weil goes on to say that Fermat would have recognised that Bachet's argument is essentially Euclid's algorithm.
_[31] 서적 , cited in All of the following citations from Fermat's ''Varia Opera'' are taken from The standard Tannery & Henry work includes a revision of Fermat's posthumous ''Varia Opera Mathematica'' originally prepared by his son
_[32] 서적 Up to the second half of the seventeenth century, academic positions were very rare, and most mathematicians and scientists earned their living in some other way (There were already some recognisable features of professional ''practice'', viz., seeking correspondents, visiting foreign colleagues, building private libraries Matters started to shift in the late 17th century ; scientific academies were founded in England (the [[Royal Society]], 1662) and France (the [[French Academy of Sciences|Académie des sciences]], 1666) and [[Russian Academy of Sciences|Russia]] (1724). Euler was offered a position at this last one in 1726; he accepted, arriving in St. Petersburg in 1727 ( and
_[33] 서적 and
_[34] 서적 and
_[35] 서적 Euler was generous in giving credit to others , not always correctly.
_[36] 서적 Varadarajan 2006
_[37] 웹사이트 Andrew Wiles on Solving Fermat https://www.pbs.org/[...] WGBH 2000-11
_[38] 서적 Disquisitiones Arithmeticae 2007
_[39] 서적 2007
_[40] 서적 2000
_[41] 서적 2004
_[42] 서적 2004
_[43] 서적 2008
_[44] 서적 2004
_[45] 문서 Brun sieve
_[46] 서적 2004
_[47] 서적 2008
_[48] 서적 2007
_[49] 문서 solvable group
_[50] 문서
_[51] 서적 Mathematical Developments Arising from Hilbert Problems American Mathematical Society
_[52] 서적 The Unreasonable Effectiveness of Number Theory American Mathematical Soc. 1992
_[53] 간행물 Computer science and its relation to mathematics The American Mathematical Monthly 1974
_[54] 서적 Applications of number theory to numerical analysis Springer-Verlag 1981
_[55] 서적 An Introduction to Number Theory with Cryptography https://www.taylorfr[...] Chapman and Hall/CRC 2018
_[56] 서적 Digital Signal Processing Algorithms : Number Theory, Convolution, Fast Fourier Transforms, and Applications https://www.worldcat[...] 2017
_[57] 논문 Physics of the Riemann Hypothesis
_[58] 서적 Error-Correcting Codes: A Mathematical Introduction https://www.taylorfr[...] Routledge 2018
_[59] 간행물 Communication Networks and Hilbert Modular Forms http://link.springer[...] Springer Netherlands 2001
_[60] 논문 Aesthetics, Dynamics, and Musical Scales: A Golden Connection http://dx.doi.org/10[...] 2002-03-01
_[61] 웹사이트 Fibonacci http://www-history.m[...] University of St Andrews
_[62] 서적 Squaring the circle-thinking the unthinkable https://books.google[...]