Let $n$ and $k$ be integers.
Need to solve $n! + 3n = k^2$, where $n!$ denotes $n$ factorial.
I do not have any ideas about this equation, except I suppose the only $6$ roots are $(0,1), (0, -1), (1, 2), (1, -2), (4, 6),$ and $(4, -6)$
Can anybody prove that these are the only roots?
Just a partial answer for now.
We can write our equation as: $$n\cdot\left((n-1)!+3\right)=k^2,$$ so, assuming that there is a prime $p$ such that $p\mid n$ and $\nu_p(n)\equiv{1}\pmod{2}$, then $p$ must divide $(n-1)!+3$. This can happen only if $p=3$. So we have only two chances: $n$ is a square or $3,27,243,\ldots$ times a square. The problems now boils down to finding the integers $m$ for which $$ m! + 3 = k^2\quad\text{or}\quad m!+3=3k^2. $$ The first case cannot occur if $m\geq 4$, since in such a case $m!+3\equiv 3\pmod{4}$ cannot be a square. If $m\geq 6$, the second case is equivalent to: $$2\cdot 4\cdot5\cdot\ldots\cdot m = (k-1)(k+1),$$ or (since $k$ must be odd) to: $$2\cdot5\cdot\ldots\cdot m = k(k+1).$$
Just a list; I put in a factoring command with a bound, so it is fast, here I said investigate prime factors of $n! + 3 n$ only up to 10,000. As you can see, the prime factorization of $n$ influences the prime factors of $n! + 3 n.$ As others have said, you can also use conditions $\bmod 4$ without knowing the full factorization of anything. Note how, for $n \geq 7,$ if $n$ is not divisible by $3$ then "OUTCOME" is divisible by $3$ but not $9,$ so cannot be a square. Oh, for the big number, I had it write cdot in case I wanted to convert to Latex later. Annoying.
Alright, read through a printout, it just support's Jack's idea that the number is ruled out by either $3$ or by a prime dividing $n$ unless $n = 3 k^2.$
1 = 1 OUTCOME = 2^2
2 = 2 OUTCOME = 2^3
3 = 3 OUTCOME = 3 cdot 5
4 = 2^2 OUTCOME = 2^2 cdot 3^2
5 = 5 OUTCOME = 3^3 cdot 5
6 = 2 * 3 OUTCOME = 2 cdot 3^2 cdot 41
7 = 7 OUTCOME = 3 cdot 7 cdot 241
8 = 2^3 OUTCOME = 2^3 cdot 3 cdot 41^2
9 = 3^2 OUTCOME = 3^3 cdot mbox{BIG}
10 = 2 * 5 OUTCOME = 2 cdot 3 cdot 5 cdot 73 cdot 1657
11 = 11 OUTCOME = 3 cdot 11 cdot 17 cdot mbox{BIG}
12 = 2^2 * 3 OUTCOME = 2^2 cdot 3^2 cdot mbox{BIG}
13 = 13 OUTCOME = 3 cdot 13 cdot 103 cdot mbox{BIG}
14 = 2 * 7 OUTCOME = 2 cdot 3 cdot 7 cdot 19 cdot 67 cdot 1021 cdot 1597
15 = 3 * 5 OUTCOME = 3^2 cdot 5 cdot 43 cdot mbox{BIG}
16 = 2^4 OUTCOME = 2^4 cdot 3 cdot 19 cdot 89 cdot mbox{BIG}
17 = 17 OUTCOME = 3 cdot 17 cdot 821 cdot 3089 cdot mbox{BIG}
18 = 2 * 3^2 OUTCOME = 2 cdot 3^3 cdot mbox{BIG}
19 = 19 OUTCOME = 3 cdot 19 cdot 47 cdot 89 cdot mbox{BIG}
20 = 2^2 * 5 OUTCOME = 2^2 cdot 3 cdot 5 cdot 37 cdot 467 cdot mbox{BIG}
21 = 3 * 7 OUTCOME = 3^2 cdot 7 cdot 29 cdot mbox{BIG}
22 = 2 * 11 OUTCOME = 2 cdot 3 cdot 11 cdot mbox{BIG}
23 = 23 OUTCOME = 3 cdot 23 cdot 283 cdot mbox{BIG}
24 = 2^3 * 3 OUTCOME = 2^3 cdot 3^2 cdot mbox{BIG}
25 = 5^2 OUTCOME = 3 cdot 5^2 cdot mbox{BIG}
26 = 2 * 13 OUTCOME = 2 cdot 3 cdot 13 cdot mbox{BIG}
27 = 3^3 OUTCOME = 3^4 cdot 37 cdot mbox{BIG}
28 = 2^2 * 7 OUTCOME = 2^2 cdot 3 cdot 7 cdot 1873 cdot mbox{BIG}
29 = 29 OUTCOME = 3 cdot 29 cdot 2011 cdot mbox{BIG}
30 = 2 * 3 * 5 OUTCOME = 2 cdot 3^2 cdot 5 cdot 331 cdot 2939 cdot mbox{BIG}
31 = 31 OUTCOME = 3 cdot 31 cdot mbox{BIG}
32 = 2^5 OUTCOME = 2^5 cdot 3 cdot 73 cdot 311 cdot 5197 cdot mbox{BIG}
33 = 3 * 11 OUTCOME = 3^2 cdot 11 cdot 157 cdot 317 cdot mbox{BIG}
34 = 2 * 17 OUTCOME = 2 cdot 3 cdot 17 cdot mbox{BIG}
35 = 5 * 7 OUTCOME = 3 cdot 5 cdot 7 cdot 43 cdot mbox{BIG}
36 = 2^2 * 3^2 OUTCOME = 2^2 cdot 3^3 cdot 251 cdot mbox{BIG}
37 = 37 OUTCOME = 3 cdot 37 cdot mbox{BIG}
38 = 2 * 19 OUTCOME = 2 cdot 3 cdot 19 cdot 163 cdot mbox{BIG}
39 = 3 * 13 OUTCOME = 3^2 cdot 13 cdot mbox{BIG}
40 = 2^3 * 5 OUTCOME = 2^3 cdot 3 cdot 5 cdot mbox{BIG}
41 = 41 OUTCOME = 3 cdot 41 cdot 109 cdot 1483 cdot mbox{BIG}
42 = 2 * 3 * 7 OUTCOME = 2 cdot 3^2 cdot 7 cdot mbox{BIG}
43 = 43 OUTCOME = 3 cdot 43 cdot mbox{BIG}
44 = 2^2 * 11 OUTCOME = 2^2 cdot 3 cdot 11 cdot mbox{BIG}
45 = 3^2 * 5 OUTCOME = 3^3 cdot 5 cdot 2293 cdot mbox{BIG}
46 = 2 * 23 OUTCOME = 2 cdot 3 cdot 23 cdot 127 cdot mbox{BIG}
47 = 47 OUTCOME = 3 cdot 47 cdot 97 cdot 967 cdot mbox{BIG}
48 = 2^4 * 3 OUTCOME = 2^4 cdot 3^2 cdot 89 cdot 137 cdot mbox{BIG}
49 = 7^2 OUTCOME = 3 cdot 7^2 cdot mbox{BIG}
50 = 2 * 5^2 OUTCOME = 2 cdot 3 cdot 5^2 cdot 149 cdot mbox{BIG}
51 = 3 * 17 OUTCOME = 3^2 cdot 17 cdot 131 cdot mbox{BIG}
52 = 2^2 * 13 OUTCOME = 2^2 cdot 3 cdot 13 cdot 4219 cdot mbox{BIG}
53 = 53 OUTCOME = 3 cdot 53 cdot 347 cdot 1489 cdot 7253 cdot mbox{BIG}
54 = 2 * 3^3 OUTCOME = 2 cdot 3^4 cdot 4339 cdot mbox{BIG}
55 = 5 * 11 OUTCOME = 3 cdot 5 cdot 11 cdot mbox{BIG}
56 = 2^3 * 7 OUTCOME = 2^3 cdot 3 cdot 7 cdot 593 cdot mbox{BIG}
57 = 3 * 19 OUTCOME = 3^2 cdot 19 cdot 71 cdot 223 cdot mbox{BIG}
58 = 2 * 29 OUTCOME = 2 cdot 3 cdot 29 cdot mbox{BIG}
59 = 59 OUTCOME = 3 cdot 59 cdot mbox{BIG}
60 = 2^2 * 3 * 5 OUTCOME = 2^2 cdot 3^2 cdot 5 cdot 1039 cdot mbox{BIG}
61 = 61 OUTCOME = 3 cdot 61 cdot 211 cdot mbox{BIG}
62 = 2 * 31 OUTCOME = 2 cdot 3 cdot 31 cdot mbox{BIG}
63 = 3^2 * 7 OUTCOME = 3^3 cdot 7 cdot 1787 cdot mbox{BIG}
64 = 2^6 OUTCOME = 2^6 cdot 3 cdot mbox{BIG}
65 = 5 * 13 OUTCOME = 3 cdot 5 cdot 13 cdot mbox{BIG}
66 = 2 * 3 * 11 OUTCOME = 2 cdot 3^2 cdot 11 cdot mbox{BIG}
67 = 67 OUTCOME = 3 cdot 67 cdot 71 cdot mbox{BIG}
68 = 2^2 * 17 OUTCOME = 2^2 cdot 3 cdot 17 cdot 8753 cdot mbox{BIG}
69 = 3 * 23 OUTCOME = 3^2 cdot 23 cdot 733 cdot mbox{BIG}
70 = 2 * 5 * 7 OUTCOME = 2 cdot 3 cdot 5 cdot 7 cdot mbox{BIG}
71 = 71 OUTCOME = 3 cdot 71 cdot 953 cdot mbox{BIG}
72 = 2^3 * 3^2 OUTCOME = 2^3 cdot 3^3 cdot 1601 cdot mbox{BIG}
73 = 73 OUTCOME = 3 cdot 73 cdot mbox{BIG}
74 = 2 * 37 OUTCOME = 2 cdot 3 cdot 37 cdot mbox{BIG}
75 = 3 * 5^2 OUTCOME = 3^2 cdot 5^2 cdot 199 cdot 211 cdot 373 cdot mbox{BIG}
76 = 2^2 * 19 OUTCOME = 2^2 cdot 3 cdot 19 cdot mbox{BIG}
77 = 7 * 11 OUTCOME = 3 cdot 7 cdot 11 cdot mbox{BIG}
78 = 2 * 3 * 13 OUTCOME = 2 cdot 3^2 cdot 13 cdot mbox{BIG}
79 = 79 OUTCOME = 3 cdot 79 cdot 113 cdot mbox{BIG}
80 = 2^4 * 5 OUTCOME = 2^4 cdot 3 cdot 5 cdot 191 cdot 4583 cdot mbox{BIG}
81 = 3^4 OUTCOME = 3^5 cdot mbox{BIG}
82 = 2 * 41 OUTCOME = 2 cdot 3 cdot 41 cdot 139 cdot mbox{BIG}
83 = 83 OUTCOME = 3 cdot 83 cdot mbox{BIG}
84 = 2^2 * 3 * 7 OUTCOME = 2^2 cdot 3^2 cdot 7 cdot mbox{BIG}
85 = 5 * 17 OUTCOME = 3 cdot 5 cdot 17 cdot mbox{BIG}
86 = 2 * 43 OUTCOME = 2 cdot 3 cdot 43 cdot mbox{BIG}
87 = 3 * 29 OUTCOME = 3^2 cdot 29 cdot mbox{BIG}
88 = 2^3 * 11 OUTCOME = 2^3 cdot 3 cdot 11 cdot 211 cdot 4639 cdot mbox{BIG}
89 = 89 OUTCOME = 3 cdot 89 cdot 97 cdot mbox{BIG}
90 = 2 * 3^2 * 5 OUTCOME = 2 cdot 3^3 cdot 5 cdot 1373 cdot mbox{BIG}
91 = 7 * 13 OUTCOME = 3 cdot 7 cdot 13 cdot 1571 cdot mbox{BIG}
92 = 2^2 * 23 OUTCOME = 2^2 cdot 3 cdot 23 cdot mbox{BIG}
93 = 3 * 31 OUTCOME = 3^2 cdot 31 cdot mbox{BIG}
94 = 2 * 47 OUTCOME = 2 cdot 3 cdot 47 cdot mbox{BIG}
95 = 5 * 19 OUTCOME = 3 cdot 5 cdot 19 cdot mbox{BIG}
96 = 2^5 * 3 OUTCOME = 2^5 cdot 3^2 cdot 179 cdot mbox{BIG}
97 = 97 OUTCOME = 3 cdot 97 cdot mbox{BIG}
98 = 2 * 7^2 OUTCOME = 2 cdot 3 cdot 7^2 cdot mbox{BIG}
99 = 3^2 * 11 OUTCOME = 3^3 cdot 11 cdot mbox{BIG}
100 = 2^2 * 5^2 OUTCOME = 2^2 cdot 3 cdot 5^2 cdot mbox{BIG}
101 = 101 OUTCOME = 3 cdot 101 cdot 1031 cdot 1277 cdot 2999 cdot mbox{BIG}
102 = 2 * 3 * 17 OUTCOME = 2 cdot 3^2 cdot 17 cdot 7933 cdot mbox{BIG}
103 = 103 OUTCOME = 3 cdot 103 cdot mbox{BIG}
104 = 2^3 * 13 OUTCOME = 2^3 cdot 3 cdot 13 cdot mbox{BIG}
105 = 3 * 5 * 7 OUTCOME = 3^2 cdot 5 cdot 7 cdot 3677 cdot mbox{BIG}
106 = 2 * 53 OUTCOME = 2 cdot 3 cdot 53 cdot 5569 cdot mbox{BIG}
107 = 107 OUTCOME = 3 cdot 107 cdot 2393 cdot mbox{BIG}
108 = 2^2 * 3^3 OUTCOME = 2^2 cdot 3^4 cdot 829 cdot mbox{BIG}
109 = 109 OUTCOME = 3 cdot 109 cdot 173 cdot 2287 cdot mbox{BIG}
110 = 2 * 5 * 11 OUTCOME = 2 cdot 3 cdot 5 cdot 11 cdot mbox{BIG}
111 = 3 * 37 OUTCOME = 3^2 cdot 37 cdot mbox{BIG}
112 = 2^4 * 7 OUTCOME = 2^4 cdot 3 cdot 7 cdot 173 cdot mbox{BIG}
113 = 113 OUTCOME = 3 cdot 113 cdot mbox{BIG}
114 = 2 * 3 * 19 OUTCOME = 2 cdot 3^2 cdot 19 cdot mbox{BIG}
115 = 5 * 23 OUTCOME = 3 cdot 5 cdot 23 cdot mbox{BIG}
116 = 2^2 * 29 OUTCOME = 2^2 cdot 3 cdot 29 cdot 1051 cdot 3617 cdot mbox{BIG}
117 = 3^2 * 13 OUTCOME = 3^3 cdot 13 cdot 1753 cdot mbox{BIG}
118 = 2 * 59 OUTCOME = 2 cdot 3 cdot 59 cdot mbox{BIG}
119 = 7 * 17 OUTCOME = 3 cdot 7 cdot 17 cdot 1439 cdot mbox{BIG}
120 = 2^3 * 3 * 5 OUTCOME = 2^3 cdot 3^2 cdot 5 cdot 149 cdot 4049 cdot mbox{BIG}
121 = 11^2 OUTCOME = 3 cdot 11^2 cdot 127 cdot mbox{BIG}
122 = 2 * 61 OUTCOME = 2 cdot 3 cdot 61 cdot mbox{BIG}
123 = 3 * 41 OUTCOME = 3^2 cdot 41 cdot 5179 cdot mbox{BIG}
124 = 2^2 * 31 OUTCOME = 2^2 cdot 3 cdot 31 cdot 857 cdot 1789 cdot mbox{BIG}
125 = 5^3 OUTCOME = 3 cdot 5^3 cdot mbox{BIG}
126 = 2 * 3^2 * 7 OUTCOME = 2 cdot 3^3 cdot 7 cdot 997 cdot 2243 cdot mbox{BIG}
127 = 127 OUTCOME = 3 cdot 127 cdot 1187 cdot 2153 cdot mbox{BIG}
128 = 2^7 OUTCOME = 2^7 cdot 3 cdot 569 cdot mbox{BIG}
129 = 3 * 43 OUTCOME = 3^2 cdot 43 cdot mbox{BIG}
130 = 2 * 5 * 13 OUTCOME = 2 cdot 3 cdot 5 cdot 13 cdot mbox{BIG}
131 = 131 OUTCOME = 3 cdot 131 cdot 191 cdot mbox{BIG}
132 = 2^2 * 3 * 11 OUTCOME = 2^2 cdot 3^2 cdot 11 cdot mbox{BIG}
133 = 7 * 19 OUTCOME = 3 cdot 7 cdot 19 cdot 743 cdot mbox{BIG}
134 = 2 * 67 OUTCOME = 2 cdot 3 cdot 67 cdot 373 cdot 577 cdot mbox{BIG}
135 = 3^3 * 5 OUTCOME = 3^4 cdot 5 cdot mbox{BIG}
136 = 2^3 * 17 OUTCOME = 2^3 cdot 3 cdot 17 cdot 1889 cdot mbox{BIG}
137 = 137 OUTCOME = 3 cdot 137 cdot 7907 cdot mbox{BIG}
138 = 2 * 3 * 23 OUTCOME = 2 cdot 3^2 cdot 23 cdot mbox{BIG}
139 = 139 OUTCOME = 3 cdot 139 cdot 6133 cdot mbox{BIG}
140 = 2^2 * 5 * 7 OUTCOME = 2^2 cdot 3 cdot 5 cdot 7 cdot mbox{BIG}
141 = 3 * 47 OUTCOME = 3^2 cdot 47 cdot mbox{BIG}
142 = 2 * 71 OUTCOME = 2 cdot 3 cdot 71 cdot mbox{BIG}
143 = 11 * 13 OUTCOME = 3 cdot 11 cdot 13 cdot mbox{BIG}
144 = 2^4 * 3^2 OUTCOME = 2^4 cdot 3^3 cdot mbox{BIG}
145 = 5 * 29 OUTCOME = 3 cdot 5 cdot 29 cdot mbox{BIG}
146 = 2 * 73 OUTCOME = 2 cdot 3 cdot 73 cdot 2659 cdot mbox{BIG}
147 = 3 * 7^2 OUTCOME = 3^2 cdot 7^2 cdot 179 cdot mbox{BIG}
148 = 2^2 * 37 OUTCOME = 2^2 cdot 3 cdot 37 cdot 211 cdot 271 cdot 2677 cdot mbox{BIG}
149 = 149 OUTCOME = 3 cdot 149 cdot mbox{BIG}
150 = 2 * 3 * 5^2 OUTCOME = 2 cdot 3^2 cdot 5^2 cdot mbox{BIG}
151 = 151 OUTCOME = 3 cdot 151 cdot mbox{BIG}
152 = 2^3 * 19 OUTCOME = 2^3 cdot 3 cdot 19 cdot 1213 cdot mbox{BIG}
153 = 3^2 * 17 OUTCOME = 3^3 cdot 17 cdot 1289 cdot mbox{BIG}
154 = 2 * 7 * 11 OUTCOME = 2 cdot 3 cdot 7 cdot 11 cdot 827 cdot 3463 cdot mbox{BIG}
155 = 5 * 31 OUTCOME = 3 cdot 5 cdot 31 cdot mbox{BIG}
156 = 2^2 * 3 * 13 OUTCOME = 2^2 cdot 3^2 cdot 13 cdot mbox{BIG}
157 = 157 OUTCOME = 3 cdot 157 cdot 379 cdot mbox{BIG}
158 = 2 * 79 OUTCOME = 2 cdot 3 cdot 79 cdot 6037 cdot mbox{BIG}
159 = 3 * 53 OUTCOME = 3^2 cdot 53 cdot 1733 cdot 5407 cdot mbox{BIG}
160 = 2^5 * 5 OUTCOME = 2^5 cdot 3 cdot 5 cdot 4003 cdot mbox{BIG}
161 = 7 * 23 OUTCOME = 3 cdot 7 cdot 23 cdot mbox{BIG}
162 = 2 * 3^4 OUTCOME = 2 cdot 3^5 cdot 331 cdot mbox{BIG}
163 = 163 OUTCOME = 3 cdot 163 cdot mbox{BIG}
164 = 2^2 * 41 OUTCOME = 2^2 cdot 3 cdot 41 cdot 271 cdot mbox{BIG}
165 = 3 * 5 * 11 OUTCOME = 3^2 cdot 5 cdot 11 cdot mbox{BIG}
166 = 2 * 83 OUTCOME = 2 cdot 3 cdot 83 cdot 239 cdot mbox{BIG}
167 = 167 OUTCOME = 3 cdot 167 cdot mbox{BIG}
168 = 2^3 * 3 * 7 OUTCOME = 2^3 cdot 3^2 cdot 7 cdot mbox{BIG}
169 = 13^2 OUTCOME = 3 cdot 13^2 cdot 373 cdot 439 cdot 1489 cdot 4799 cdot mbox{BIG}
170 = 2 * 5 * 17 OUTCOME = 2 cdot 3 cdot 5 cdot 17 cdot mbox{BIG}
171 = 3^2 * 19 OUTCOME = 3^3 cdot 19 cdot 317 cdot mbox{BIG}
172 = 2^2 * 43 OUTCOME = 2^2 cdot 3 cdot 43 cdot mbox{BIG}
173 = 173 OUTCOME = 3 cdot 173 cdot mbox{BIG}
174 = 2 * 3 * 29 OUTCOME = 2 cdot 3^2 cdot 29 cdot 883 cdot 3539 cdot mbox{BIG}
175 = 5^2 * 7 OUTCOME = 3 cdot 5^2 cdot 7 cdot 211 cdot mbox{BIG}
176 = 2^4 * 11 OUTCOME = 2^4 cdot 3 cdot 11 cdot 6899 cdot mbox{BIG}
177 = 3 * 59 OUTCOME = 3^2 cdot 59 cdot mbox{BIG}
178 = 2 * 89 OUTCOME = 2 cdot 3 cdot 89 cdot mbox{BIG}
179 = 179 OUTCOME = 3 cdot 179 cdot 239 cdot 3593 cdot mbox{BIG}
180 = 2^2 * 3^2 * 5 OUTCOME = 2^2 cdot 3^3 cdot 5 cdot mbox{BIG}
181 = 181 OUTCOME = 3 cdot 181 cdot 653 cdot mbox{BIG}
182 = 2 * 7 * 13 OUTCOME = 2 cdot 3 cdot 7 cdot 13 cdot 5581 cdot mbox{BIG}
183 = 3 * 61 OUTCOME = 3^2 cdot 61 cdot 337 cdot 5749 cdot mbox{BIG}
184 = 2^3 * 23 OUTCOME = 2^3 cdot 3 cdot 23 cdot mbox{BIG}
185 = 5 * 37 OUTCOME = 3 cdot 5 cdot 37 cdot 571 cdot mbox{BIG}
186 = 2 * 3 * 31 OUTCOME = 2 cdot 3^2 cdot 31 cdot 1901 cdot mbox{BIG}
187 = 11 * 17 OUTCOME = 3 cdot 11 cdot 17 cdot mbox{BIG}
188 = 2^2 * 47 OUTCOME = 2^2 cdot 3 cdot 47 cdot mbox{BIG}
189 = 3^3 * 7 OUTCOME = 3^4 cdot 7 cdot 3089 cdot mbox{BIG}
190 = 2 * 5 * 19 OUTCOME = 2 cdot 3 cdot 5 cdot 19 cdot mbox{BIG}
191 = 191 OUTCOME = 3 cdot 191 cdot mbox{BIG}
192 = 2^6 * 3 OUTCOME = 2^6 cdot 3^2 cdot 4973 cdot mbox{BIG}
193 = 193 OUTCOME = 3 cdot 193 cdot mbox{BIG}
194 = 2 * 97 OUTCOME = 2 cdot 3 cdot 97 cdot 839 cdot mbox{BIG}
195 = 3 * 5 * 13 OUTCOME = 3^2 cdot 5 cdot 13 cdot mbox{BIG}
196 = 2^2 * 7^2 OUTCOME = 2^2 cdot 3 cdot 7^2 cdot 947 cdot mbox{BIG}
197 = 197 OUTCOME = 3 cdot 197 cdot 421 cdot mbox{BIG}
198 = 2 * 3^2 * 11 OUTCOME = 2 cdot 3^3 cdot 11 cdot 5653 cdot mbox{BIG}
199 = 199 OUTCOME = 3 cdot 199 cdot 761 cdot 1259 cdot mbox{BIG}
200 = 2^3 * 5^2 OUTCOME = 2^3 cdot 3 cdot 5^2 cdot 433 cdot mbox{BIG}
jagy@phobeusjunior:~$
