Topology

Intersection of complete funcoids ★★

Author(s): Porton

Conjecture   If $ f $, $ g $ are complete funcoids (generalized closures) then $ f \cap^{\mathsf{\tmop{FCD}}} g $ is a complete funcoid (generalized closure).

Keywords: complete funcoid; funcoid; generalized closure

Posted by porton
updated September 2nd, 2007
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Number Theory » Computational N.T.

Counterexamples to the Baillie-PSW primality test ★★

Author(s):

Problem  (1)   Find a counterexample to Baillie-PSW primality test or prove that there is no one.
Problem  (2)   Find a composite $ n\equiv 3 $ or $ 7\pmod{10} $ which divides both $ 2^{n-1} - 1 $ (see Fermat pseudoprime) and the Fibonacci number $ F_{n+1} $ (see Lucas pseudoprime), or prove that there is no such $ n $.

Keywords:

Posted by maxal
updated August 7th, 2007
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Number Theory » Computational N.T.

A sextic counterexample to Euler's sum of powers conjecture ★★

Author(s): Euler

Problem   Find six positive integers $ x_1, x_2, \dots, x_6 $ such that $$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$ or prove that such integers do not exist.

Keywords:

Posted by maxal
updated August 5th, 2007
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Number Theory » Combinatorial N.T.

Divisibility of central binomial coefficients ★★

Author(s): Graham

Problem  (1)   Prove that there exist infinitely many positive integers $ n $ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7) = 1.$$
Problem  (2)   Prove that there exists only a finite number of positive integers $ n $ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7\cdot 11) = 1.$$

Keywords:

Posted by maxal
updated June 29th, 2014
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