Make your own free website on Tripod.com

Solutions to Znám's Problem

A finite set of integers of length k is a solution to Znám's problem when each integer in the set is a proper divisor of 1 plus the product of the other integers in the set.

Solutions for k = 5

2, 3, 7, 47, 395
2, 3, 11, 23, 31

Solutions for k = 6

2, 3, 7, 43, 1823, 193667
2, 3, 7, 47, 403, 19403
2, 3, 7, 47, 415, 8111
2, 3, 7, 47, 583, 1223
2, 3, 7, 55, 179, 24323

Solutions for k = 7

2, 3, 7, 43, 1807, 3263447, 2130014000915
2, 3, 7, 43, 1807, 3263591, 71480133827
2, 3, 7, 43, 1807, 3264187, 14298637519
2, 3, 7, 43, 3263, 4051, 255895 1
2, 3, 7, 43, 3559, 3667, 33816127
2, 3, 7, 47, 395, 779831, 6020372531
2, 3, 7, 67, 187, 283, 334651
2, 3, 11, 17, 101, 149, 3109
2, 3, 11, 23, 31, 47063, 442938131
2, 3, 11, 23, 31, 47095, 59897203
2, 3, 11, 23, 31, 47131, 30382063
2, 3, 11, 23, 31, 47243, 12017087
2, 3, 11, 23, 31, 47423, 6114059
2, 3, 11, 23, 31, 49759, 866923
2, 3, 11, 23, 31, 60563, 211031
2, 3, 11, 25, 29, 1097, 2753
2, 3, 11, 31, 35, 67, 369067
2, 3, 13, 25, 29, 67, 2981

Solutions for k = 8

2, 3, 7, 43, 1807, 3263443, 10650057792155, 134811739261383753719
2, 3, 7, 43, 1807, 3263443, 10652778201539, 41691378583707695
2, 3, 7, 43, 1807, 3263443, 10699597306267, 2300171639909623
2, 3, 7, 43, 1807, 3263447, 2130014387399, 11739058070963394487
2, 3, 7, 43, 1807, 3263479, 288182779055, 243811701792623
2, 3, 7, 43, 1807, 3263483, 260604226747, 80249212735823
2, 3, 7, 43, 1807, 3263495, 200947673239, 67137380077902268343
2, 3, 7, 43, 1807, 3263495, 200949404503, 23316080984691 959
2, 3, 7, 43, 1807, 3263531, 119666789791, 8081907028348841339
2, 3, 7, 43, 1807, 3263779, 31834629787, 4396910340967
2, 3, 7, 43, 1807, 3316627, 203509259, 109643149191047
2, 3, 7, 43, 1807, 3586039, 36800447, 2550097247
2, 3, 7, 43, 1811, 655519, 389313431, 1507818475
2, 3, 7, 43, 1811, 713899, 7813583, 2409102303622951
2, 3, 7, 43, 1811, 793595, 3722287, 233296531681207
2, 3, 7, 43, 1817, 298637, 279594269, 3859101523354821017
2, 3, 7, 43, 1819, 252731, 2134319143, 6047845668256680791
2, 3, 7, 43, 1823, 193667, 637617223459, 312735 17203328870463055
2, 3, 7, 43, 1823, 193675, 46832109 19, 754794584867
2, 3, 7, 43, 1831, 132347, 231679879, 1197240789041771
2, 3, 7, 43, 1891, 40379, 9444811, 55866875
2, 3, 7, 43, 1943, 25615, 456729463, 450222796871
2, 3, 7, 43, 1951, 30571, 118463, 14484098803019
2, 3, 7, 43, 2105, 12773, 2775277, 168100338289
2, 3, 7, 43, 2137, 16921, 37501, 49708999789
2, 3, 7, 43, 2755, 5407, 172771, 357538828973647
2, 3, 7, 43, 2813, 5045, 692705317, 188433744928309
2, 3, 7, 47, 395, 779731, 607979652647, 21743485766025360000683
2, 3, 7, 47, 395, 779731, 607979652683, 6974325623477705424647
2, 3, 7, 47, 395, 779731, 607979653531, 410254449012081168631
2, 3, 7, 47, 395, 779731, 607979655287, 139119028839856004123
2, 3, 7, 47, 395, 779731, 607979697799, 8183472856913555659
2, 3, 7, 47, 395, 779731, 607979793451, 2624887933109395 11 1
2, 3, 7, 47, 395, 779731, 607982046587, 154405744751990423
2, 3, 7, 47, 395, 779743, 46768385339, 1672627310178141725483
2, 3, 7, 47, 395, 779747, 35764242947, 12154487527525118239
2, 3, 7, 47, 395, 779827, 6286857907, 2158880732959
2, 3, 7, 47, 395, 781727, 305967719, 125881309327
2, 3, 7, 47, 395, 7821 11, 257276179, 57278664659
2, 3, 7, 47, 395, 782287, 277442411, 1701723083
2, 3, 7, 47, 395, 782611, 211810259, 1592773460578079
2, 3, 7, 47, 395, 816247, 17428931, 652510750371360683
2, 3, 7, 47, 395, 1108727, 2627707, 140495574531059
2, 3, 7, 47, 403, 19403, 15435513395, 8215692183434294399
2, 3, 7, 47, 403, 19403, 15435513463, 2456237880094942747
2, 3, 7, 47, 403, 19403, 15435516179, 84697872837562655
2, 3, 7, 47, 415, 8111, 6644612339, 1522443894582665279
2, 3, 7, 47, 415, 8111, 6644613463, 38292177286592827
2, 3, 7, 47, 415, 8111, 6644645747, 1320426321921983
2, 3, 7, 47, 449, 4477, 12137, 34035763385
2, 3, 7, 47, 583, 1223, 1407479807, 48317057302587443
2, 3, 7, 47, 583, 1223, 1468268915, 33995520959
2, 3, 7, 47, 583, 1223, 2202310039, 3899834875
2, 3, 7, 53, 209, 10589, 19651, 86321
2, 3, 7, 53, 269, 817, 7301713, 48932949591475
2, 3, 7, 53, 401, 409, 351691, 397617853
2, 3, 7, 55, 179, 24323, 10057317287, 5949978284730273323
2, 3, 7, 55, 179, 24323, 10057317311, 2467064172726591731
2, 3, 7, 55, 179, 24323, 10057317467, 513449911932648503
2, 3, 7, 55, 179, 24323, 10057317967, 145121431390804003
2, 3, 7, 55, 179, 24323, 10057320619, 30202945461748519
2, 3, 7, 55, 179, 24323, 10057325347, 12523178395739983
2, 3, 7, 55, 179, 24323, 10057454579, 736667018400959
2, 3, 7, 61, 187, 485, 150809, 971259409
2, 3, 7, 61, 293, 457, 551, 21709309
2, 3, 7, 65, 121, 6271, 1579937, 2869621
2, 3, 7, 71, 103, 65059, 1101031, 4400294969594807
2, 3, 11, 17, 79, 301, 1049, 3696653
2, 3, 11, 17, 97, 151, 444161, 317361415625
2, 3, 11, 23, 31, 47059, 2214502427, 980804197623275639
2, 3, 11, 23, 31, 47059, 2214502475, 92528699894575367
2, 3, 11, 23, 31, 47059, 2214502687, 18505741750517011
2, 3, 11, 23, 31, 47059, 2214502831, 11990273552017987
2, 3, 11, 23, 31, 47059, 2214504467, 2398056482005535
2, 3, 11, 23, 31, 47059, 2214524099, 226233749172527
2, 3, 11, 23, 31, 47059, 2214610807, 45248521436443
2, 3, 11, 23, 31, 47059, 2215070383, 8636647107907
2, 3, 11, 23, 31, 47059, 2217342227, 1729101023519
2, 3, 11, 23, 31, 47059, 2244604355, 165128325167
2, 3, 11, 23, 31, 47059, 2294166883, 63772955407
2, 3, 11, 23, 31, 47059, 2365012087, 34797266971
2, 3, 11, 23, 31, 47059, 2446798471, 23325584587
2, 3, 11, 23, 31, 47059, 2612824727, 14526193019
2, 3, 11, 23, 31, 47059, 3375982667, 6436718855
2, 3, 11, 23, 31, 47063, 447473399, 43702604167
2, 3, 11, 23, 31, 47119, 36349891, 4619150372467
2, 3, 11, 23, 31, 47147, 24928579, 11061526082145911
2, 3, 11, 23, 31, 47479, 5307047, 2371471764522551
2, 3, 11, 23, 31, 47491, 5161279, 4952592862147
2, 3, 11, 23, 31, 74963, 126415, 259118345891
2, 3, 11, 23, 31, 84527, 106159, 84453127154999
2, 3, 11, 25, 29, 787, 264841, 2542873
2, 5, 7, 11, 17, 157, 961, 4398619

Solutions for k = 9

These are the publically known solutions. To my knowledge, an exhaustive search has yet to be carried out, though it is known that all the solutions for this length begin with 2 and the vast majority begin 2, 3, ...

2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000447, 2572987736655734348107429290411162753668127385839515

Solutions for k = 10

These are the publically known solutions. To my knowledge, an exhaustive search has yet to be carried out.

2, 3, 11, 23, 31, 47059, 2214502423, 4904020979258368507, 24049421765006207593444550012151040547, 115674937446230858658157460659985774139375256845351399814552547262816571295

Solutions for k = 13

3, 4, 5, 7, 29, 41, 67, 89701, 230865947737, 5726348063558735709083, 172509500849902989281836693100308633431804359, 723726187, 4332177380360519, ?, ?, ?

Mathworld tells us that the last three numbers of this solution exceeds forty digits each, though I'm sure that it's still within Mathematica's capabilities to calculate and display these numbers. So as soon as I figure out how to calculate them based on the numbers I already know, I will put them in here.

Doublechecking the solutions

To be absolutely sure I had the correct solutions, I used the following Mathematica function, defined by Alonso Delarte:

ZnamSolQ[list_] := Module[{flag, i, max, dP1}, flag = True; i = 1; max = Length[list] + 1; While[flag && i < max, dP1 = (((Times @@ list)/list[[i]]) + 1)/list[[i]]; If[Not[IntegerQ[dP1] && dP1 > 1], flag = False]; i++]; flag]

Of course I wonder if there's a more efficient way to express the deleted product than by multiplying the whole set and then dividing the element you didn't want in the product to begin with. Also, note that this function does not check that the elements are distinct (thus, {1, 1, 1} is incorrectly flagged as a solution).

But other than these minor quibbles, the function works exactly as expected. I tried setting the attribute of the function to Listable, but that didn't work quite as I expected. Then I thought of using Notepad's Mass Replace to change the HTML table cell delimiters to "ZnamSolQ[{" and "}]" and pasting that into Mathematica (with linebreaks between each command). In a breeze, Mathematica spat out a long column of Trues.

References