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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 .
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." https://www.math.wayne.edu/ugresearch/egyfra.html.
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