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A twin Pythagorean triple is a Pythagorean triple (a,b,c) for which two values are consecutive integers. By definition, twin triplets are therefore primitive triples. Of the ...

There are two identities known as Catalan's identity. The first is F_n^2-F_(n+r)F_(n-r)=(-1)^(n-r)F_r^2, where F_n is a Fibonacci number. Letting r=1 gives Cassini's ...

Given any real number theta and any positive integer N, there exist integers h and k with 0<k<=N such that |ktheta-h|<1/N. A slightly weaker form of the theorem states that ...

136·2^(256) approx 1.575×10^(79). According to Eddington, the exact number of protons in the universe, where 136 was the reciprocal of the fine structure constant as best as ...

An Euler pseudoprime to the base b is a composite number n which satisfies b^((n-1)/2)=+/-1 (mod n). The first few base-2 Euler pseudoprimes are 341, 561, 1105, 1729, 1905, ...

Let F_n be the nth Fibonacci number. Then the sequence {F_n}_(n=2)^infty={1,2,3,5,8,...} is complete, even if one is restricted to subsequences in which no two consecutive ...

A pyramidal number of the form n(n+1)(5n-2)/6, The first few are 1, 8, 26, 60, 115, ... (OEIS A002413). The generating function for the heptagonal pyramidal numbers is ...

A formula for the generalized Catalan number _pd_(qi). The general formula is (n-q; k-1)=sum_(i=1)^k_pd_(qi)(n-pi; k-i), where (n; k) is a binomial coefficient, although ...

Let a graph G have exactly 2n-3 graph edges, where n is the number of graph vertices in G. Then G is "generically" rigid in R^2 iff e^'<=2n^'-3 for every subgraph of G having ...

The (upper) vertex independence number of a graph, often called simply "the" independence number, is the cardinality of the largest independent vertex set, i.e., the size of ...