Number
Theory| 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; 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, ...
@ General references: Hardy & Wright 60; in Honsberger 76; Weil 79, 84;
Hasse 80; Guy 81; Hua 82; Baker 85; Rose 88; Ireland & Rosen 90; Bunch
00 [I]; Nathanson
00 [elementary methods]; Everest & Ward 05 [II/III]; Coppel 06 [II].
@ And quantum mechanics: Benioff PRA(01)qp/00, qp/00-in, 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," re
unique prime decomposition of every integer; In the 3rd cy BC, Eratosthenes
conceived his "sieve" method for identifying prime numbers.
* Applications: Cryptography; life cycle periods of cicadas.
* Prime number theorem:
The number
of primes 
(x)
smaller or equal to x grows asymptotically like
(x) 
li
x:= 
2x dt/log t x/log x .
* Mersenne primes: The ones of the form 2n–1,
like 23–1 = 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);
Bombieri ThSc(92);
Peterman mp/00 [renormalization
group
approach]; Crandall & Pomerance 05 [computational]; 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].
@ Special topics: Kupershmidt a0806 [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-PRL [and
quantum
chaos]; Sekatskii a0709.
@ 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].
> Online resources:
The Prime Pages website.
Special Topics > s.a. types
of numbers; Euler's Totient Function.
* 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).
* 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 n2 – q,
conjectured by Euler and first proved by Gauß.
@ Perfect numbers: Davis ht/04 [odd].
@ Other topics: Olivastro ThSc(90) [Fermat], ThSc(90)
[magic squares]; Crandall
SA(97)feb [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:
Satisfies 
(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.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
27 jul 2008