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If n>1 and n|1^(n-1)+2^(n-1)+...+(n-1)^(n-1)+1, is n necessarily a prime? In other words, defining s_n=sum_(k=1)^(n-1)k^(n-1), does there exist a composite n such that s_n=-1 ...

The uniformity conjecture postulates a relationship between the syntactic length of expressions built up from the natural numbers using field operations, exponentials, and ...

For a finite group G, let p(G) be the subgroup generated by all the Sylow p-subgroups of G. If X is a projective curve in characteristic p>0, and if x_0, ..., x_t are points ...

If C_1, C_2, ...C_r are sets of positive integers and union _(i=1)^rC_i=Z^+, then some C_i contains arbitrarily long arithmetic progressions. The conjecture was proved by van ...

Also called the Tait flyping conjecture. Given two reduced alternating projections of the same knot, they are equivalent on the sphere iff they are related by a series of ...

Let n be a positive integer and r(n) the number of (not necessarily distinct) prime factors of n (with r(1)=0). Let O(m) be the number of positive integers <=m with an odd ...

Let p(n) be the first prime which follows a prime gap of n between consecutive primes. Shanks' conjecture holds that p(n)∼exp(sqrt(n)). Wolf conjectures a slightly different ...

Let gamma(G) denote the domination number of a simple graph G. Then Vizing (1963) conjectured that gamma(G)gamma(H)<=gamma(G×H), where G×H is the graph product. While the ...

Barnette's conjecture asserts that every 3-connected bipartite cubic planar graph is Hamiltonian. The only graph on nine or fewer vertices satisfying Barnette's conditions is ...

Any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal grid (i.e., the honeycomb, illustrated above). Pappus refers to the ...