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# Znám's Problem

A problem posed by the Slovak mathematician Stefan Znám in 1972 asking whether, for all integers , there exist integers all greater than 1 such that is a proper divisor of for each . The answer is negative for (Jának and Skula 1978) and affirmative for (Sun Qi 1983). Sun Qi also gave a lower bound for the number of solutions.

All solutions for have now been computed, summarized in the table below. The numbers of solutions for , 3, ... terms are 0, 0, 0, 2, 5, 15, 93, ... (OEIS A075441), and the solutions themselves are given by OEIS A075461.

 known solutions references 2 0 -- Jának and Skula (1978) 3 0 -- Jának and Skula (1978) 4 0 -- Jának and Skula (1978) 5 2 2, 3, 7, 47, 395 2, 3, 11, 23, 31 6 5 2, 3, 7, 43, 1823, 193667 2, 3, 7, 47, 403, 19403 2, 3, 7, 47, 415, 8111 2, 3, 7, 47, 583, 1223 2, 3, 7, 55, 179, 24323 7 15 2, 3, 7, 43, 1807, 3263447, 2130014000915 Jának and Skula (1978) 2, 3, 7, 43, 1807, 3263591, 71480133827 Cao, Liu, and Zhang (1987) 2, 3, 7, 43, 1807, 3264187, 14298637519 2, 3, 7, 43, 3559, 3667, 33816127 2, 3, 7, 47, 395, 779831, 6020372531 2, 3, 7, 67, 187, 283, 334651 2, 3, 11, 17, 101, 149, 3109 2, 3, 11, 23, 31, 47063, 442938131 2, 3, 11, 23, 31, 47095, 59897203 2, 3, 11, 23, 31, 47131, 30382063 2, 3, 11, 23, 31, 47243, 12017087 2, 3, 11, 23, 31, 47423, 6114059 2, 3, 11, 23, 31, 49759, 866923 2, 3, 11, 23, 31, 60563, 211031 2, 3, 11, 31, 35, 67, 369067 8 93 Brenton and Vasiliu (1998) 9 ? 2, 3, 7, 43, 1807, 3263443, Sun (1983) 10650056950807, 113423713055421844361000447, 2572987736655734348107429290411162753668127385839515 10 ? 2, 3, 11, 23, 31, 47059, Sun (1983) 2214502423, 4904020979258368507, 24049421765006207593444550012151040547, 115674937446230858658157460659985774139375256845351399814552547262816571295

Cao and Sun (1988) showed that and Cao and Jing (1998) that there are solutions for . A solution for was found by Girgensohn in 1996: 3, 4, 5, 7, 29, 41, 67, 89701, 230865947737, 5726348063558735709083, followed by large numbers having 45, 87, and 172 digits.

It has been observed that all known solutions to Znám's problem provide a decomposition of 1 as an Egyptian fraction

Conversely, every solution to this Diophantine equation is a solution to Znám's problem, unless for some .

Egyptian Fraction

This entry contributed by Margherita Barile

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## References

Brenton, L. and Jaje, L. "Perfectly Weighted Graphs." Graphs Combin. 17, 389-407, 2001.Brenton, L, and Vasiliu, A. "Znam's Problem." Math. Mag. 75, 3-11, 2002.Cao, Z. and Jing, C. "On the Number of Solutions of Znám's Problem." J. Harbin Inst. Tech. 30, 46-49, 1998.Cao, Z. and Sun, Q. "On the Equation and the Number of Solutions of Znám's Problem." Northeast. Math. J. 4, 43-48, 1988.Cao, Z.; Liu, R.; and Zhang, L. "On the Equation and Znám's Problem. J. Number Th. 27, 206-211, 1987.Jának, J. and Skula, L. "On the Integers for which Holds." Math. Slovaca 28, 305-310, 1978.Sloane, N. J. A. Sequences A075441 and A075461 in "The On-Line Encyclopedia of Integer Sequences."Sun, Q. "On a Problem of Š. Znám." Sichuan Daxue Xuebao, No. 4, 9-12, 1983.Wayne State University Undergraduate Mathematics Research Group. "The Egyptian Fraction: The Unit Fraction Equation." http://www.math.wayne.edu/ugresearch/egyfra.html.

Znám's Problem

## Cite this as:

Barile, Margherita. "Znám's Problem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ZnamsProblem.html