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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 ...

Let t be a nonnegative integer and let x_1, ..., x_t be nonzero elements of Z_p which are not necessarily distinct. Then the number of elements of Z_p that can be written as ...

The sum of powers of even divisors of a number. It is the analog of the divisor function for even divisors only and is written sigma_k^((e))(n). It is given simply in terms ...

An even perfect number is perfect number that is even, i.e., an even number n whose sum of divisors (including n itself) equals n. All known perfect numbers are even, and ...

Let A be a sum of squares of n independent normal standardized variates X_i, and suppose A=B+C where B is a quadratic form in the x_i, distributed as chi-squared with h ...

Let X be a metric space, A be a subset of X, and d a number >=0. The d-dimensional Hausdorff measure of A, H^d(A), is the infimum of positive numbers y such that for every ...

A formula for the number of Young tableaux associated with a given Ferrers diagram. In each box, write the sum of one plus the number of boxes horizontally to the right and ...

Let sigma_infty(n) be the sum of the infinitary divisors of a number n. An infinitary k-multiperfect number is a number n such that sigma_infty(n)=kn. Cohen (1990) found 13 ...

Pick a point O in the interior of a quadrilateral which is not a parallelogram. Join this point to each of the four vertices, then the locus of points O for which the sum of ...

A magic hexagram is a hexagram partitioned into triangles such that the sums of numbers in the six directions illustrated above sum to the same number. There are exactly two ...