For analogues in number fields, See 11R04
- 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
- 11A07: Congruences; primitive roots; residue systems
- 11A15: Power residues, reciprocity
- 11A25: Arithmetic functions; related numbers; inversion formulas
- 11A41: Primes
- 11A51: Factorization; primality
- 11A55: Continued fractions, For approximation results, See 11J70; See also 11K50, 30B70, 40A15
- 11A63: Radix representation; digital problems, For metric results, See 11K16
- 11A67: Other representations
- 11A99: None of the above but in this section
Parent field: 11: Number Theory.
Browse all (old) classifications for this area at the AMS.
Well-known texts with an elementary focus include:
- LeVeque, William J.: "Fundamentals of number theory", Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977, 280 pp. ISBN 0-201-04287-8
- Dudley, Underwood: "Elementary number theory", W. H. Freeman and Co., San Francisco, Calif., 1978. 249 pp. ISBN 0-7167-0076-*
- Stark, Harold M. : "An introduction to number theory", MIT Press, Cambridge, Mass.-London, 1978. 347 pp. ISBN 0-262-69060-8
- Rosen, Kenneth H.: "Elementary number theory and its applications", Addison-Wesley Publishing Co., Reading, MA, 1988. 466 pp., ISBN 0-201-11958-7
- Burton, David M. : "Elementary number theory", W. C. Brown Publishers, Dubuque, IA, 1989. 450 pp. ISBN 0-697-05919-7
A survey which perhaps illustrates the difference between "elementary" number theory and "easy" number theory is by Diamond, Harold G.: "Elementary methods in the study of the distribution of prime numbers", Bull. Amer. Math. Soc. 7 (1982) 553--589.
Online Lecture notes [X.-D. Jia]
See the references for number theory in general.
- Basic how-tos of continued fractions (applied to Pi ).
- Getting rational approximations using Farey sequences and continued fractions
- Connection between continued fractions and the Euclidean algorithm
- How well do continued fractions behave for sqrt(D) ?
- Is the period length for the continued fraction expansion of, say, sqrt(x^2-18) bounded independent of x? (no)
- Relationships between sets of numbers and their continued-fraction partial quotients (e.g. algebraic numbers, numbers with bounded quotients)
- Sets of numbers with good continued-fraction approximations (Mahler, Koksma)
- Markov's theorem relationing continued fractions and c(k)=[(k+1)x]-[kx]-[x]
- Partial quotients in continued fraction for Pi roughly follow the Gauss-Kusmin distribution
- Algorithms for manipulating continued fractions -- pointers, literature.
- HAKMEM algorithms for dealing with arithmetic with continued fractions
- Evaluating "regular" continued fractions with differential equations
- Lehmer evaluation of continued fractions with quotients in arithmetic prog
- Extension of continued fractions to complex numbers.
- Multidimensional analogues of continued fractions (summary and bibliography)
- Citations regarding multidimensional continued fractions
- Use of generalized continued fractions to simultaneously approximate several numbers by small rationals
- Multi-dimensional analogues of (i.e. simultaneous) continued fractions
- Why do the last few digits of a^n cycle?
- Bonus primes in Fermat's little theorem: when p^2 divides N^p - N.
- Small solutions to 2^n = s mod n
- Announcement: 2^756839 - 1 is prime.
- What is the correct sign in the congruence ((p-1)/2)! = +-1 mod p? (cf. Wilson's theorem). Answer: depends on class number formula.
- Proofs of infinitude of primes, and unique factorization from The Book.
- Topological proof of the infinitude of primes.
- Proof of Bertrand's postulate: there is always a prime between n and 2n.
- C implementation of the sieve of Eratosthenes.
- A formula for primes! (Mills's "M^(3^n)").
- Show there is a prime of the form k*2^n + 1 for every odd k less than 78557 (none for k=78557).
- Expressing a rational as sums of Egyptian fractions (1/n)
- Erdös's 4/n problem (write each 4/n as a sum of three Egyptian fractions)
- What numbers are sums of two Egyptian fractions?
- Background on the Erdös' 4/n problem
- Polynomial solutions to the 4/n problem
- Perfect numbers -- recent literature
- Multiply-perfect numbers (sum of divisors is a multiple of n)
- An odd perfect number? (Almost! :-)
- Amicable numbers (each the sum of the other's divisors)
- Long cycles of amicable numbers
- Sociable numbers (each the sum of the divisors of the next)
- Untouchable numbers (those not the sum of divisors of any other integer)
- Bogus proof of Collatz conjecture [For more information about the Collatz problem see this summary.]
- Striking data on the Collatz conjecture
- Crandall's conjecture (obvious generalization of the Collatz conjecture)
- Connection between Collatz and Catalan conjectures.
- Recasting the Collatz conjecture as a complex functional equation
- Magic squares -- history, literature, some solutions.
- De la Loubère's method of creating magic squares.
- Detailed description: how to create magic squares
- Magic squares -- literature review.
- A bibliography on magic squares.
- More references on magic squares.
- Counting the dimensions of magic squares and cubes.
- Dame Ollerenshaw's enumeration of certain magic squares.
- The unique magic hexagon
- A rational square of the form A.A in its decimal expansion!
- Squares which, in base 10, are written with only two or three distinct digits.
- Iterate this procedure: multiply the nonzero digits of n together to get n'; repeat until one digit remains. What digit is it?
- Multiply all the digits (even zero); repeat until single-digit.
- Given n find M so M*5^n has no zeros in its decimal expansion.
- Congruence conjecture on !n = 1! + 2! + ... + n!
- Seeking solutions to the set of congruences ab=c mod (a+b), bc=a mod (b+c), ca=b mod (c+a)
- Arrange the first few integers on a circle with modest successive differences
- Facts about primes which are "the mathematical equivalent of junk food".
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