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 One and only (Posted on 2016-04-28)
There is only one prime p such that p! is p digits long.

Find it and prove its uniqueness.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer assisted solution | Comment 3 of 4 |
`prime  length of                    factorial       factorial   2     1                                                   2    3     1                                                   6   5     3                                                 120   7     4                                                5040  11     8                                            39916800  13    10                                          6227020800  17    15                                     355687428096000  19    18                                  121645100408832000  23    23                             25852016738884976640000  29    31                     8841761993739701954543616000000  31    34                  8222838654177922817725562880000000  37    44        13763753091226345046315979581580902400000000  41    50  33452526613163807108170062053440751665152000000000   43    53  47    60  53    70  59    81  61    84  67    95  71   102  73   106  79   117  83   125  89   137  97   152 101   160 `

The only match on the table is for p=23, which has a 23-digit factorial.

After 100, each successive factorial's length goes up by at least 2 for every increase of the number by 1, so no numbers, prime or not, beyond 100 will have a factorial whose length matches the number.

More fundamentally, now that I think about it, even after 29, each successive number's factorial's length increased by at least 1, and the factorial's length was already greater than the number itself -- no need to go all the way to over 100.

In fact, only 1, 22, 23 and 24 have the property that each one's factorial's length is equal to the number itself.  Of these, only 23 is prime.

Table generated by:

5   open "1andonly.txt" for output as #2
10   repeat
20    P=nxtprm(P)
30    Pf=!(P)
40    Ps\$=cutspc(str(Pf))
50    L=len(Ps\$)
60    print #2,P,L
70   until P>100
80   close #2

supplemented by a variation also printing Pf, the actual factorial.

 Posted by Charlie on 2016-04-28 10:34:13

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