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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
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E-mail address : pdg@worldofnumbers.com