Number Theory

In General > s.a. mathematics.
* Idea: The study of the operations + and ×, usually on integers.
* History: Contributors were Euclid, Diophantus; Fermat, Euler, Lagrange, Legendre, Fourier, Gauss, Cauchy, Abel, Jacobi, Dirichlet, Liouville; Kummer, Galois, Hermite, Eisenstein, Kronecker, Riemann, Dedekind, Bachmann, Gordan, H Weber, G Cantor, Hurwitz, Minkowski.
* Fermat's first theorem: If p is a prime, and a any integer, p divides either a or ap−1 − 1; Proved by Fermat.
* Fermat's second theorem: The numbers 2n+1 are prime; It is wrong for n ≥ 6.
* Open problems: For example, the Goldbach and Langlands conjectures; > see conjectures.
* Nice fact: The sequence [(51/2+1)/2]m approaches an integer as m → ∞.
* Nice fact: Given any 10 numbers between 1 and 100, there are always two pairs whose sums are equal; Likewise for 20 numbers between 1 and 5000. (Claimed not to be too difficult to prove.)
* Conjecture: Take any integer, n1; If it is even, divide by 2, n2 = n1/2, and if it is odd, n2 = 3n1+1; Iterate; Then it is thought that eventually the iteration becomes periodic: ..., 4, 2, 1, 4, 2, 1, ...
@ Simple introductions: Bunch 00; Duverney 10 [elementary intro through Diophantine equations]; Forman & Rash 15.
@ General references: Hardy & Wright 60; in Honsberger 76; Weil 79, 84; Hasse 80; Hua 82; Narkiewicz 84; Baker 85; Ireland & Rosen 90; Rose 94; Goldman 97 [historical]; Nathanson 99 [elementary methods]; Guy 04 [unsolved problems]; Andreescu et al 06 [problems]; Coppel 09 [II]; Everest & Ward 10 [II/III]; Li et al 13 [and applications]; Jarvis 14 [algebraic].
@ And quantum mechanics: Benioff PRA(01)qp/00, qp/00-proc, Algo(02)qp/01, a0704 [quantum representations of numbers]; Tran AP(04) [partitions and many-particle density of states].

Prime Numbers
* History: 350 BC, Euclid's "Fundamental Theorem of Arithmetic," about the unique prime decomposition of every integer; In the 3rd century BC, Eratosthenes conceived his "sieve" method for identifying prime numbers; GIMPS, The Great Internet Mersenne Prime Search, a collaborative effort to find large primes using many computers; 2015, The Electronic Frontier Foundation has awards fo people who find very large primes.
* Applications: Cryptography; The life-cycle periods of cicadas.
* Prime number theorem: The number of primes π(x) smaller or equal to x grows asymptotically like

π(x) ~ li x:= 2x dt/log t ~ x/log x .

* Mersenne primes: The ones of the form 2n−1, like 23−1 = 7; 2018, the largest known prime number is $$2^{77,232,917}-1$$, with 23,249,425 digits.
* Double Wieferich Primes: 2000, The only known ones are p = 2, q = 1093; p = 3, q = 1006003 ; p = 5, q = 1645333507; p = 83, q = 4871; p = 911, q = 318917; p = 2903, q = 18787.
@ General references: Ribenboim 91; Olivastro ThSc(90)may; Bombieri ThSc(92)sep; Peterman mp/00 [renormalization-group approach]; Gepner m.NT/05 [distribution]; Granville BAMS(05) [determining whether a number is prime]; Muñoz & Pérez CMP(08); Green & Tao AM(08) [primes contain aribtrarily long arithmetic progressions]; Crandall & Pomerance 10 [computational]; news ns(13)mar [and quantum computers]; Mazur & Stein 16 [and the Riemann hypothesis]; news sn(18)jan [the largest known prime].
@ Special topics: Kupershmidt a0806-wd [Nicolas conjecture / inequality].
@ Differences: Kumar et al cm/03 [distribution]; Ares & Castro PhyA(06)cm/03; Szpiro PhyA(04), PhyA(07) [gaps].
@ As spectrum of quantum H: Mussardo cm/97; Rosu MPLA(03); Timberlake & Tucker a0708/PhyA [and quantum chaos]; Sekatskii a0709; Menezes & Svaiter a1211 [no-go result].
@ Other physics: Liboff & Wong IJTP(98) [quasi-chaos in sequence]; Gadiyar & Padma ht/98 [prime pairs and quantum field theory]; Kelly & Pilling ht/01 [twin and triplet primes]; Bonanno & Mega CSF(04) [dynamical approach]; Latorre & Sierra QIC-a1302 [and pure quantum states], a1403 [entanglement in the primes].
> Online resources: The Prime Pages website.

Other Special Numbers > s.a. types of numbers.
* Triangular numbers: A number is triangular if it is half the sum of two consecutive integers; Every positive integer is the sum of 3 triangular numbers (Gauss).
* Perfect numbers: Numbers which are equal to the sum of their factors; The first five are 6, 28, 496, 8128, and 33,550,336; For each Mersenne prime 2n−1, there is a perfect number 2n−1 (2n−1), like 22 (23−1) = 28, or 21257786 (21257787−1).
* Figurate numbers: Numbers that can be represented by regular geometrical arrangements of equally spaced points; They include triangular numbers, square numbers, pentagonal numbers, and other polygonal numbers.
@ References: Davis ht/04 [odd perfect numbers]; Deza & Deza 12 [figurate numbers].

Special Topics > s.a. Euler's Totient Function; knot theory [arithmetic topology]; partitions.
* Elliptic curves and modular forms: STW (Shimura-Taniyama-Weil) conjecture, proved in 1999 [@ news NAMS(99)dec], after A Wiles proved a special case in his proof of Fermat's last theorem; It is part of the Langlands program.
* Quadratic reciprocity theorem: A result on the form of the prime divisors p of numbers of the form n2q, conjectured by Euler and first proved by Gauß.
@ Factoring numbers: Clauser & Dowling PRA(96)-a0810 [using Young's N-slit interferometer]; Altschuler & Williams a1402 [simulated annealing approach]; Dridi & Alghassi sRep(17)-a1604 [using quantum annealing and computational algebraic geometry].
@ Other topics: Olivastro ThSc(90)may [Fermat], ThSc(90)nov [magic squares]; Crandall SA(97)feb [manipulating large numbers and computers].

Geometric Number Theory
* History: Not just a branch of number theory; It is now independent, with many applications and connections.
* Typical problems: All related to properties of lattices in En and bases, the dense ball packing problem, the Minkowski-Hlawka theorem, etc, and can range to reduction of polynomials or coding.
* Measure on the space of lattices: It satisfies μ(total) = 1.
* Topology on the space of lattices: A sequence Ln converges to L if the bases converge, vector by vector.
* Classification: Bravais types; Types of dual tilings.