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;
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;
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:= ∫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.
* 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 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];
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
n2 − 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 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.
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send feedback and suggestions to bombelli at olemiss.edu – modified 26 apr 2021