Is $k=553276187$ the smallest solution

elementary-number-theorynumber theoryprime factorization

I search the smallest positive integer $k$, such that $40!+k$ splits into three primes having $16$ decimal digits. The smallest solution I found is

$$k=553\ 276\ 187$$

You can see the factorization here : http://factordb.com/index.php?id=1100000001197449481

According to my calculations, no $k\le 2\cdot 10^6$ does the job.

Is my $k$ the optimal solution ?

Best Answer

The answer to the question is no, here is a slightly smaller solution: $$ 494\;804\;473\ . $$

sage: factor( factorial(40) + 494804473 )
8912658466556113 * 9232286052422441 * 9915818101022081

It was obtained also in sage, the program was running one day, and this is in this intermediate situation a found solution. (It is still running. It was not written to touch the numbers one by one starting with $40!$ in order, so this may also not be the minimal number doing the job.)

Note: As i also said in the comment, the fact that $40!^{1/3}$ is so close to the number $999\dots 9$ with $16$ digits makes it harder to get such decompositions with three factors of $16,16,16$ digits each. Because the range of the primes is not really between the $16$ digits numbers $1000\dots 0$ and $999\dots 9$. The computer was founding for instance also

k=    33377 40!+k = 1056334373719901 * 22413347482811693 * 34461718591042889
k=    42269 40!+k = 1859373333290887 * 2719883435120179 * 161334845671400153
k=    47189 40!+k = 1822739844251111 * 15851904710438411 * 28238324719193609
k=    55049 40!+k = 1101465899839957 * 3378301824756127 * 219268155834100091

with factors having $\ge 16$ digits, and also insisting that the smaller prime factor among all three of the three is bigger than $7000\dots 0$ ($16$ digits)... 

k=300722963 40!+k = 7073009263847467 * 8126384986908847 * 14195263159044887
k=349245887 40!+k = 7209029739646403 * 8613116168531887 * 13140380656971067
k= 77639477 40!+k = 7252468942583363 * 7992027613367573 * 14076744068584523
k=300731429 40!+k = 7321097013439289 * 7404841114350401 * 15050577417998861
k=339499973 40!+k = 7398940260351053 * 10008305130239719 * 11018310576981439
k= 96999373 40!+k = 7532964349652891 * 9232067350535819 * 11732219831109637
k= 48531323 40!+k = 7564931816003599 * 10225749107742383 * 10547387873068219
k=465611681 40!+k = 7866432514968563 * 7912332084219151 * 13108794049712437

k=465608309 40!+k = 8221566296360779 * 9835813811956651 * 10089745359967421
k=126205139 40!+k = 8278202201940731 * 9790182376796351 * 10067421590328119
k=388112077 40!+k = 8601556182258047 * 8687613919887803 * 10918614312712297
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