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[ December 3, 2012 ]
Pandigital PRP's of the form abcd! +/- efghij
A 4-digit factorial and a 6-digit displacement.

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.

 1024! + 865937 1025! + 978643 1039! + 684527 1058! + 643927 1064! + 295387 1072! + 548693 1208! + 657439 1208! + 749653 1237! + 468509 1249! + 560873 1256! + 348097 1267! + 408953 1268! + 903457 1295! + 368047 1409! + 583267 1463! + 805297 1468! + 205937 1478! + 965023 1486! + 530279 1538! + 762049 1538! + 942607 1564! + 930827 1580! + 672349 1604! + 259837 1609! + 825347 1642! + 857039 1652! + 807493 1750! + 829463 1768! + 205493 1805! + 673429 1825! + 640973 1840! + 726953 1894! + 326057 2015! + 483697 2035! + 748169 2063! + 548719 2086! + 547139 2086! + 759431 2087! + 615493 2095! + 816743 2165! + 340897 2354! + 810697 2486! + 375019 2518! + 306479 2546! + 983701 2569! + 410387 2659! + 704183 2675! + 438091 2678! + 103549 2684! + 571093 2690! + 351847 2690! + 743851 2710! + 568349 2896! + 570413 2906! + 875341 2915! + 608347 2945! + 837601 2957! + 403861 2960! + 174583 3052! + 681497 3526! + 907481 3574! + 810269 3640! + 528719 3815! + 764209 3865! + 970421 3965! + 214087 4012! + 583697 4087! + 356129 4093! + 257861 4237! + 608591 4607! + 235891 4670! + 582391 4682! + 701359 4802! + 659173 4805! + 237691 4829! + 176503 4895! + 271603 4910! + 528673 4985! + 610327 5347! + 689201 5380! + 241679 5710! + 968423 5806! + 197243 5906! + 128347 6014! + 253789 6095! + 184273 6104! + 283957 6184! + 302759 6271! + 450839 6502! + 734189 6835! + 104729 6850! + 342791 6985! + 270143 7015! + 948263 7168! + 530249 7261! + 458309 7268! + 903451 7528! + 436091 7540! + 863921 7618! + 450239 8206! + 159437 8294! + 306157 8362! + 159407 8527! + 149603 8734! + 596021 8975! + 402631 9031! + 268547 1028! - 465739 1048! - 362759 1048! - 659237 1052! - 643879 1063! - 254987 1072! - 498653 1096! - 523847 1204! - 879653 1204! - 936587 1268! - 593407 1270! - 584963 1304! - 869257 1342! - 680597 1432! - 589607 1459! - 678203 1468! - 523907 1507! - 426389 1508! - 763429 1582! - 490367 1670! - 348259 1685! - 432907 1702! - 583469 1832! - 640957 1867! - 590243 2015! - 684379 2018! - 765439 2056! - 871349 2183! - 695407 2390! - 547681 2459! - 103867 2485! - 106397 2495! - 867301 2531! - 409867 2543! - 160879 2564! - 138079 2584! - 136709 2609! - 475381 2689! - 470531 2701! - 985463 2738! - 450691 2759! - 608431 2851! - 637409 2930! - 586471 2960! - 785143 3046! - 217859 3062! - 147859 3098! - 642517 3280! - 459167 3406! - 982571 3421! - 658079 3460! - 192587 3470! - 651289 3520! - 768941 3572! - 468109 3586! - 192047 3781! - 264059 3790! - 845261 3985! - 276041 4057! - 398621 4216! - 578309 4310! - 756289 4367! - 258019 4603! - 297581 4706! - 581239 4709! - 128563 4837! - 216509 4856! - 709231 5032! - 741869 5047! - 812639 5239! - 618407 5290! - 861347 5438! - 276019 5471! - 283609 5702! - 136849 5728! - 903641 5780! - 921643 5792! - 864301 5974! - 610823 6028! - 734159 6035! - 784129 6037! - 182549 6145! - 238709 6190! - 875243 6254! - 738109 6347! - 518209 6428! - 350971 6458! - 213097 6470! - 583291 6485! - 137209 6485! - 203971 6491! - 782053 6508! - 273149 6508! - 973421 6850! - 423179 7298! - 643051 7408! - 162593 7520! - 968431 7594! - 620813 7942! - 305861 7946! - 302851 8024! - 756139 8027! - 143569 8062! - 473159 8246! - 390751 8432! - 610579 8596! - 240173 8635! - 217409 8905! - 217643 9034! - 856721 9065! - 413827 9461! - 520837 9725! - 380641 9820! - 617453 9860! - 715423

Underlined factorials! have pandigital solutions
on both the positive and the negative displacement side.
( Ninedigital version of this topic please consult page ninedigits.htm )

Note that 7408! - 162593 has a palindromic digitlength of 25452 !
Check it out for instance with PFGW using the following command
pfgw64 -f0 -od -q"len(7408!-162593)"
though the last minus-part may be discarded.

Or by using WolframAlpha.
Just type in the inputbox 7408!-162593

A000181 Prime Curios! Prime Puzzle
Wikipedia 181 Le nombre 181
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