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 *a*^{p–1}–1; Proved by Fermat.

* __Fermat's second theorem__:
The numbers 2^{n}+1
are prime; wrong for *n* ≥ 6.

* __Open problems__: For
example, the Goldbach and Langlands conjectures; > see conjectures.

* __Nice fact__: The sequence
[(5^{1/2}+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, *n*_{1};
If it is even, divide by 2, *n*_{2} = *n*_{1}/2,
and if it is odd, *n*_{2} = 3*n*_{1}+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*:= ∫_{2}^{x} d*t*/log *t* ~ *x*/log *x* .

* __Mersenne primes__: The ones of the form 2^{n}–1,
like 2^{3}–1 = 7.

* __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].

@ __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 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 2^{n}–1,
there is a perfect number 2^{n–1}(2^{n}–1),
like 2^{2}(2^{3}–1)
= 28, or 2^{1257786 }(2^{1257787}–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].

> __Online resources__: see Wikipedia page on square-free
integers; MathWorld page and Wikipedia page on 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 *n*^{2} – *q*,
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 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 E^{n} 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__: Satisfies *μ*(total) = 1.

* __Topology on the space of
lattices__:
A sequence *L*_{n} converges to *L* if the bases converge, vector by vector.

* __Classification__: Bravais
types; Types of dual tilings.

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send feedback and suggestions to bombelli at olemiss.edu – modified
17 jun 2016