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;
It is 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;
Kowalski 21 [probabilistic].

@ __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;
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.

* __Semiprime numbers__:
Natural numbers that are products of two prime numbers.

@ __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 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];
news SA(20)oct [the number 42 :-)].

> __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 sRep(17)-a1604 [using quantum annealing and computational algebraic geometry];
Cadavid et al a2104
[using diffusion as a computational engine].

@ __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__:
It 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 26 apr 2021