Znám's Problem

Znám's problem, posed by the Slovak mathematician Stefan Znám in 1972 (Janák and Skula 1978), asks 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 (Janák and Skula 1978) and affirmative for (Sun 1983). Here counts solutions up to permutation of the entries. Sun (1983) also gave a lower bound for .

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, 18, 96, ... (OEIS A075441), and the solutions themselves are given by OEIS A075461.

		known solutions	references
2	0	--	Janák and Skula (1978)
3	0	--	Janák and Skula (1978)
4	0	--	Janák 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	18	2, 3, 7, 43, 1807, 3263447, 2130014000915	Janák 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
		2, 3, 7, 43, 3263, 4051, 2558951
		2, 3, 11, 25, 29, 1097, 2753
		2, 3, 13, 25, 29, 67, 2981
8	96		OEIS A075441
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

Sun and Cao (1988) showed that and Cao and Jing (1998) that there are solutions for . A solution for was found by Girgensohn in 1996 (Brenton and Vasiliu 2002, p. 8): 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 .