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 
.
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.
