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If bc=bd (mod a) and (b,a)=1 (i.e., a and b are relatively prime), then c=d (mod a).

A subfield which is strictly smaller than the field in which it is contained. The field of rationals Q is a proper subfield of the field of real numbers R which, in turn, is ...

If R is a ring (commutative with 1), the height of a prime ideal p is defined as the supremum of all n so that there is a chain p_0 subset ...p_(n-1) subset p_n=p where all ...

A composite knot is a knot that is not a prime knot. Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition is ...

An algorithm for making tables of primes. Sequentially write down the integers from 2 to the highest number n you wish to include in the table. Cross out all numbers >2 which ...

Consider the Euler product zeta(s)=product_(k=1)^infty1/(1-1/(p_k^s)), (1) where zeta(s) is the Riemann zeta function and p_k is the kth prime. zeta(1)=infty, but taking the ...

A set S of positive integers is said to be Diophantine iff there exists a polynomial Q with integral coefficients in m>=1 indeterminates such that ...

The probability that a random integer between 1 and x will have its greatest prime factor <=x^alpha approaches a limiting value F(alpha) as x->infty, where F(alpha)=1 for ...

There are several types of numbers that are commonly termed "lucky numbers." The first is the lucky numbers of Euler. The second is obtained by writing out all odd numbers: ...

A negative number multiplied by another negative number gives a positive number.