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Znám's problem (Definition)

Given a length $ k$, is it possible to construct a set of integers $ n_1, \ldots , n_k$ such that each

$\displaystyle n_i \vert(1 + \prod_{j \ne i}^n n_j)$

as a proper divisor? This is Znám's problem.

This problem has solutions for $ k > 4$, and all solutions for $ 4 < k < 9$ have been found, and a few for higher $ k$ are known. The Sylvester sequence provides many of the solutions. At Wayne State University in 2001, Brenton and Vasiliu devised an algorithm to exhaustively search for solutions for a given length, and thus they found all solutions for $ k = 8$. Their algorithm, though smarter than a brute force search, is still computationally intense the larger $ k$ gets.

Solutions to the problem have applications in continued fractions and perfectly weighted graphs.

The problem is believed to have been first posed by Štefan Znám in 1972. Qi Sun proved in 1983 that there are solutions for all $ k > 4$.

References

Brenton, L, and Vasiliu, A. “Znam's Problem." Math. Mag. 75, 3-11, 2002.



"Znám's problem" is owned by Mravinci.
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Other names:  Znam's problem

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a few examples of solutions to Znám's problem (Example) by PrimeFan
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Cross-references: References, graphs, continued fractions, applications, force, algorithm, Sylvester sequence, solutions, proper divisor, integers, length
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This is version 5 of Znám's problem, born on 2006-03-21, modified 2006-10-17.
Object id is 7755, canonical name is ZnamsProblem.
Accessed 1842 times total.

Classification:
AMS MSC: 11A55 (Number theory :: Elementary number theory :: Continued fractions)
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