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The Diophantine equation D x ² + 2 2 m + 1 = y

J. H. E. Cohn (2003)

Colloquium Mathematicae

It is shown that for a given squarefree positive integer D, the equation of the title has no solutions in integers x > 0, m > 0, n ≥ 3 and y odd, nor unless D ≡ 14 (mod 16) in integers x > 0, m = 0, n ≥ 3, y > 0, provided in each case that n does not divide the class number of the imaginary quadratic field containing √(-2D), except for a small number of (stated) exceptions.

The distribution of the sum-of-digits function

Michael Drmota, Johannes Gajdosik (1998)

Journal de théorie des nombres de Bordeaux

By using a generating function approach it is shown that the sum-of-digits function (related to specific finite and infinite linear recurrences) satisfies a central limit theorem. Additionally a local limit theorem is derived.

The Josephus problem

Lorenz Halbeisen, Norbert Hungerbühler (1997)

Journal de théorie des nombres de Bordeaux

We give explicit non-recursive formulas to compute the Josephus-numbers j ( n , 2 , i ) and j ( n , 3 , i ) and explicit upper and lower bounds for j ( n , k , i ) (where k 4 ) which differ by 2 k - 2 (for k = 4 the bounds are even better). Furthermore we present a new fast algorithm to calculate j ( n , k , i ) which is based upon the mentioned bounds.

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