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Znám's Problem

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A problem posed by the Slovak mathematician Stefan Znám in 1972 asking whether, for all integers k>=2, there exist k integers x_1,...,x_k all greater than 1 such that x_i is a proper divisor of x_1...x_k/x_i+1 for each i=1,...,k. The answer is negative for 2<=k<=4 (Jának and Skula 1978) and affirmative for k>=5 (Sun Qi 1983). Sun Qi also gave a lower bound for the number Z(k) of solutions.

All solutions for 5<=k<=8 have now been computed, summarized in the table below. The numbers of solutions for n=2, 3, ... terms are 0, 0, 0, 2, 5, 15, 93, ... (OEIS A075441), and the solutions themselves are given by OEIS A075461.

kZ(k)known solutions x_1,...,x_kreferences
20--Jának and Skula (1978)
30--Jának and Skula (1978)
40--Jának and Skula (1978)
522, 3, 7, 47, 395
2, 3, 11, 23, 31
652, 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
7152, 3, 7, 43, 1807, 3263447, 2130014000915Jának and Skula (1978)
2, 3, 7, 43, 1807, 3263591, 71480133827Cao, 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
893Brenton 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 Z(11)>=5 and Cao and Jing (1998) that there are >=39 solutions for n>=12. A solution for k=13 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

1/(x_1)+1/(x_2)+...+1/(x_k)+1/(x_1...x_k)=1.

Conversely, every solution to this Diophantine equation is a solution to Znám's problem, unless x_i=x_1...x_k/x_i+1 for some i.

See also

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 sum_(j=1)^(s)1/x_1...x_s=n 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 sum_(j=1)^(s)(1/x_j)+(1/(x_1...x_s))=1 and Znám's Problem. J. Number Th. 27, 206-211, 1987.Jának, J. and Skula, L. "On the Integers x_i for which x_i|x_1...x_(i-1)x_i...x_n+1 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.

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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

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