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A test for the primality of Fermat numbers F_n=2^(2^n)+1, with n>=2 and k>=2. Then the two following conditions are equivalent: 1. F_n is prime and (k/F_n)=-1, where (n/k) is ...

Pickover's sequence gives the starting positions in the decimal expansion of pi (ignoring the leading 3) in which the first n digits of e occur (counting the leading 2). So, ...

Write the exact powers of 2 and 3 in sorted order as 1, 2, 3, 4, 8, 9, 16, 27, 32, ... (OEIS A006899), and let u_n be the nth term in the sequence. Then u_(n+1)-u_n tends to ...

The Pippenger product is an unexpected Wallis-like formula for e given by e/2=(2/1)^(1/2)(2/34/3)^(1/4)(4/56/56/78/7)^(1/8)... (1) (OEIS A084148 and A084149; Pippenger 1980). ...

The prime distance pd(n) of a nonnegative integer n is the absolute difference between n and the nearest prime. It is therefore true that pd(p)=0 for primes p. The first few ...

Let a!=b, A, and B denote positive integers satisfying (a,b)=1 (A,B)=1, (i.e., both pairs are relatively prime), and suppose every prime p=B (mod A) with (p,2ab)=1 is ...

The base-4 method of counting in which only the digits 0, 1, 2, and 3 are used. The illustration above shows the numbers 0 to 63 represented in quaternary, and the following ...

The sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ... (OEIS A002024) consisting of 1 copy of 1, 2 copies of 2, 3 copies of 3, and so on. Surprisingly, there exist simple formulas ...

The silver ratio is the quantity defined by the continued fraction delta_S = [2,2,2,...] (1) = 2+1/(2+1/(2+1/(2+...))) (2) (Wall 1948, p. 24). It follows that ...

The continued fraction for mu is given by [1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...] (OEIS A099803). The positions at which the numbers 1, 2, ... occur in the continued ...