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[ December 3, 2012 ] The (probable) primes from the next list of pandigitals (split into 4-digit factorials and a displacement of the remaining 6 digits) must have prime displacements (both positive and negative). ps. the same is true for the ninedigital case (resp. 4 and 5 digits) (also known as 'zeroless pandigital'). Suppose we have a composite displacement then we could split it into its factors. One of the factors of this 6-digit number is always smaller than the factorial number itself and hence we can extract a common factor: [ 1*2*3*4*...*(n-1)*(n) ] +/- [ f1*...*fn ] [ cf ] * [ [ 1*2*3*4*...*(n-1)*(n) ] +/- [ f1*...*fn ] ] rendering the whole shebang divisible by this common factor yielding no longer a (probable) prime but a composite. Hence the 6-digit displacement itself must be prime as well ! A smallest composite in this (probable) prime expression must be a 7-digit displacement.
Underlined factorials! have pandigital solutions
Note that 7408! - 162593 has a palindromic digitlength of 25452 ! Check it out for instance with PFGW using the following command
Or by using WolframAlpha.
For reference goals and easy searching all the nine- & pandigitals
implicitly displayed in these topics are listed here.
Topic → 1024865937, 1025978643, 1039684527, 1058643927, 1064295387, 1072548693, | |||||
A000181 Prime Curios! Prime Puzzle Wikipedia 181 Le nombre 181 |
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