Factorials
Posted: October 8th, 2021, 8:55 pm
This puzzle is to find all solutions of
n! * (n-1)! = m!
where m and n are non-negative integers.
Cinelli
n! * (n-1)! = m!
where m and n are non-negative integers.
Cinelli
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NotSure wrote:I am probably just embarrassing myself here, but since m! = m * (m-1)!, then are the only solutions where m! = 1 or (m-1)! = 1, i.e. m = n = 1 or 2?
NotSure wrote:I am probably just embarrassing myself here, ...
UncleEbenezer wrote:If Gengulphus hasn't solved it, I infer it's not easy. I wonder if it's not merely not easy, but unsolved/unsolvable?
cinelli wrote:This puzzle is to find all solutions of
n! * (n-1)! = m!
where m and n are non-negative integers.
Cinelli
....| 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 |
----+----+----+----+----+----+----+----+----+----+----+----+----
2 | 1 | | | | | | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
3 | 1 | 1 | | | | | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
4 | 3 | 1 | | | | | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
5 | 3 | 1 | 1 | | | | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
6 | 4 | 2 | 1 | | | | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
7 | 4 | 2 | 1 | 1 | | | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
8 | 7 | 2 | 1 | 1 | | | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
9 | 7 | 4 | 1 | 1 | | | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
10 | 8 | 4 | 2 | 1 | | | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
11 | 8 | 4 | 2 | 1 | 1 | | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
12 | 10 | 5 | 2 | 1 | 1 | | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
13 | 10 | 5 | 2 | 1 | 1 | 1 | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
14 | 11 | 5 | 2 | 2 | 1 | 1 | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
15 | 11 | 6 | 3 | 2 | 1 | 1 | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
16 | 15 | 6 | 3 | 2 | 1 | 1 | | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
17 | 15 | 6 | 3 | 2 | 1 | 1 | 1 | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
18 | 16 | 8 | 3 | 2 | 1 | 1 | 1 | | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
19 | 16 | 8 | 3 | 2 | 1 | 1 | 1 | 1 | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
20 | 18 | 8 | 4 | 2 | 1 | 1 | 1 | 1 | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
21 | 18 | 9 | 4 | 3 | 1 | 1 | 1 | 1 | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
22 | 19 | 9 | 4 | 3 | 2 | 1 | 1 | 1 | | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
23 | 19 | 9 | 4 | 3 | 2 | 1 | 1 | 1 | 1 | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
24 | 22 | 10 | 4 | 3 | 2 | 1 | 1 | 1 | 1 | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
25 | 22 | 10 | 6 | 3 | 2 | 1 | 1 | 1 | 1 | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
26 | 23 | 10 | 6 | 3 | 2 | 2 | 1 | 1 | 1 | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
27 | 23 | 13 | 6 | 3 | 2 | 2 | 1 | 1 | 1 | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
28 | 25 | 13 | 6 | 4 | 2 | 2 | 1 | 1 | 1 | | |
----+----+----+----+----+----+----+----+----+----+----+----+----
29 | 25 | 13 | 6 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | |
----+----+----+----+----+----+----+----+----+----+----+----+----
30 | 26 | 14 | 7 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | |
----+----+----+----+----+----+----+----+----+----+----+----+----
31 | 26 | 14 | 7 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 1 |
----+----+----+----+----+----+----+----+----+----+----+----+----
32 | 31 | 14 | 7 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 1 |
----+----+----+----+----+----+----+----+----+----+----+----+----
| | | | | | | | | | | |
NotSure wrote:When my research led me to papers by Paul 'another roof, another proof' Erdos, I quickly realised that I was not merely out of my depth, but at the bottom of the Mariana Trench.
Lootman wrote:Perhaps you should be grateful. Paul Erdos had a life of tragedy and was dead by the age of 33.
Lootman wrote:NotSure wrote:When my research led me to papers by Paul 'another roof, another proof' Erdos, I quickly realised that I was not merely out of my depth, but at the bottom of the Mariana Trench.
Perhaps you should be grateful. Paul Erdos had a life of tragedy and was dead by the age of 33.
The smartest kid at my school was a child prodigy at maths. He had his A level in pure maths by age 15 and, since he could not go up to Trinity College, Cambridge for another 2/3 years on grounds of age, did a B.Sc. in maths in his spare time at school just for something to do. I believe he had a Ph.D by the time he was 21. He self reported his IQ as 182 and the guy was incapable of lying so I can believe it.
He was also the most socially awkward kid I ever knew, with zero social skills. He was dead at age 26, suicide I heard. I guess the world was just too stupid for him.
NotSure wrote:Lootman wrote:NotSure wrote:When my research led me to papers by Paul 'another roof, another proof' Erdos, I quickly realised that I was not merely out of my depth, but at the bottom of the Mariana Trench.
Perhaps you should be grateful. Paul Erdos had a life of tragedy and was dead by the age of 33.
The smartest kid at my school was a child prodigy at maths. He had his A level in pure maths by age 15 and, since he could not go up to Trinity College, Cambridge for another 2/3 years on grounds of age, did a B.Sc. in maths in his spare time at school just for something to do. I believe he had a Ph.D by the time he was 21. He self reported his IQ as 182 and the guy was incapable of lying so I can believe it.
He was also the most socially awkward kid I ever knew, with zero social skills. He was dead at age 26, suicide I heard. I guess the world was just too stupid for him.
Erdos lived to the ripe old age of 83 but was, by all accounts, quite eccentric. He too obtained a PhD at 21. The problem above can (IMHO) be framed as combinatorics, so if anyone could crack it, I suspect he could. It seems many exceptional mathematicians have done their best work by their mid 20's. It sounds like the prodigy you knew had little else to fall back on. A sad story.
UncleEbenezer wrote:If Gengulphus hasn't solved it, I infer it's not easy. I wonder if it's not merely not easy, but unsolved/unsolvable?
n b lower limit for primes p definitely in
m (= 2n-(2b-1)) n+1 <= p <= m range
==============================================
1 1 1 none
2 2 1 none
3 2 3 none
4 3 3 none
5 3 5 none
6 3 7 7
7 3 9 none
8 4 9 none
9 4 11 11
10 4 13 11,13
11 4 15 13
12 4 17 13,17
13 4 19 17,19
14 4 21 17,19
15 4 23 17,19,23
16 5 23 17,19,23
17 5 25 19,23
18 5 27 19,23
19 5 29 23,29
20 5 31 23,29,31
21 5 33 23,29,31
22 5 35 23,29,31
23 5 37 29,31,37
24 5 39 29,31,37
25 5 41 29,31,37,41
26 5 43 29,31,37,41,43
27 5 45 29,31,37,41,43
28 5 47 29,31,37,41,43,47
29 5 49 31,37,41,43,47
30 5 51 31,37,41,43,47
31 5 53 37,41,43,47,53