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abundant number – mutually prime
abundant number
A natural number whose distinct divisors, including 1 but not including itself, have a sum greater than the number. For example, the proper factors of 12 are 1, 2, 3, 4, and 6, which sum to 16, so 12 is abundant. A number whose proper factors sum to less than the number is called deficient, and one whose factors sum to precisely the number is called perfect.
 Cf. perfect number.
algebraic number
A real or complex number which is the root of a polynomial with rational coefficients. Numbers that cannot be so expressed are called transcendental numbers. The algebraic numbers are countable.
Carmichael number
A composite integer n which satisfies the congruence an = a modulo n for all integers a.
Thus they pass the Fermat test without being prime.
Carmichael numbers are characterised by the Korselt condition that they are composite, squarefree and p-1 | n-1 for every prime factor p | n.
The first Carmichael number is 561 = 3 × 11 × 17.
It was proved in 1994 by Alford, Granville and Pomerance that there are infinitely many Carmichael numbers.
deficient number
See: abundant number.
Diophantine equation
A polynomial equation with integer coefficients. (Named after the 3rd century Greek mathematician Diophantus of Alexandria.)
Cf. Hilbert's Problems (the tenth problem), Fermat's Last Theorem.
divide
To divide a number a by another number b is to find a third number c such that the product of b and c is a, that is, b × c = a. The number a is called the dividend, the number b is called the divisor, and the number c is called the quotient. The operation of dividing may be denoted by a horizontal or diagonal slash separating the dividend and divisor (with the dividend on top), or by a horizontal dash with a dot above and below it placed between the dividend and divisor.
In the case of whole numbers a and b there may not be a whole number quotient; however, there are always unique whole numbers q and r such that a = b × q + r, with r < b. In this case q is called the quotient and r is called the remainder. If in a particular case r = 0, we say that b divides a, and this is often denoted by b|a.
e
See Euler number.
Euclid number
See perfect number.
Euler number
The transcendental number e, approximately 2.71828.... It may be defined as the limit
or as the infinite sum
The function ex is called the exponential function, and has the property that it is its own derivative.
Euler phi function
Given a natural number n, this function returns the number of integers between 1 and n which are relatively prime to n.
Fermat’s Last Theorem
The Diophantine equation x n + y n = z n has no non-trivial solutions in integers x, y, and z for any n greater than 2. (There are infinitely many solutions in integers for n = 2, and these are called the Pythagorean triples.) Fermat penned this theorem in the margin of his copy of the Arithmetica of Diophantus of Alexandria, and added, “I have discovered a remarkable proof of this theorem, which unfortunately this margin is too small to contain.” He died without ever writing down the proof, and the theorem remained unproved for 300 years, until Andrew Wiles presented a proof at a Cambridge lecture in 1992.
Fermat’s Little Theorem
For any integer n and prime number p that does not divide n, n p - 1 is congruent to 1 modulo p.
Cf. Euler phi function.
Fermat test
A test for primality based on the converse of Fermat's little theorem:
the Fermat test for n base b is that the congruence bp-1  1 modulo n should hold. For prime n, this is always provided that
b and n are coprime; however there
are also some composite numbers which will pass the test. For example, 341 = 11 × 31 is composite, yet 2340 1 modulo 341, so we call 2 a Fermat pseudoprime to the base 2.
Even worse, Carmichael numbers are pseudoprimes for the Fermat test and any base.
Fortunately, pseudoprimes and Carmichael numbers are rare compared to primes.
Such tests are thus probabilistic, in that numbers passing the test are likely but not certain to be prime.
Fibonacci sequence
ARTICLE
The sequence discovered by the medieval mathematician Fibonacci, and described in his book Liber Abaci. See the article for a complete description.
Fundamental Theorem of Arithmetic
ARTICLE
Every natural number is either prime or may be decomposed uniquely into prime factors.
g.c.d.
Abbreviation for greatest common divisor.
Gelfond-Schneider Theorem
If a is a complex number that is also algebraic, and if b is an irrational number, then a b is a transcendental number.
greatest common divisor
The greatest common divisor of two natural numbers a and b is the largest natural number c such that c divides a and c divides b.
imaginary number
By definition, the square root of –1, i.e., i 2 = –1.
Cf. complex number.
irrational number
A real number which is not a rational number, i.e., which cannot be expressed as a ratio of integers. Examples: p (the ratio of the circumference of a circle to its diameter) and the square root of 2 (ratio of the length of the diagonal of a square to the length of one side). The irrational numbers are uncountably infinite.
Related article: Irrationality of the Square Root of 2
Related MiniText: Number -- What Is How Many?
mutually prime
Two integers are mutually prime if they have no common factors larger than 1 or -1.
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