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As the title states, how would I go about finding the positive integer solutions of

$$\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}+\frac{1}{x_1 x_2 \cdots x_n}=1$$?

Thank you for your help.

Edit: I think I've made some progress. I conjecture that you can expand the set {a, a+1, a*(a+1)+1, a*(a+1)(a(a+1)+1)+1, ...} (i.e. multiply all previous solutions and add 1), where a=2, to size n, and that will always be a solution in the case of n variables. This is far from finding all positive integer solutions, though.

Edit 2: For example {2}, {2,3}, {2,3,7}, {2,3,7,43}, {2,3,7,1807}, {2,3,7,43,1807,3263443} are all solutions in the case where $n=$ size of the solution set respectively.

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3  
Taking $x_1 = \ldots = x_n$ gives no solutions, while if not all $x_i$ are the same, but two of them are not coprime, then you will also get no solutions (as then $\frac{1}{x_1} + \ldots + \frac{1}{x_n}$ has denominator smaller than $x_1 \cdot \ldots \cdot x_n$). So any solution should have all $x_i$ pairwise coprime. –  TMM Oct 1 '11 at 13:44
    
@Edit: What exactly is your solution? Is the $n$ in your solution infinite? –  TMM Oct 1 '11 at 13:49
    
I messed up a little. I meant you can expand that set to form new solutions (will edit for clarity). –  josh Oct 1 '11 at 13:51
3  
You solutions are Sylvester's sequence, oeis.org/A000058 –  Ross Millikan Oct 1 '11 at 15:08
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2 Answers

up vote 14 down vote accepted

This is known as the improper Znám problem.

All of the solutions for $n\le 7$, as well as an algorithm to compute them are given in Brenton and Hill's 1998 paper. All of the 119 solutions for $n=8$ are reported to have been found in Brenton and Vasiliu's 2002 paper, but the link containing the solutions appears to have decayed. Regenerating the first hundred or so of these solutions is easy, but finishing the last few will probably take a couple of weeks of computer time. Finding all the solutions for $n=9$ looks intractable without a breakthrough in the search techniques.

Brenton, Lawrence; Hill, Richard (1988), "On the Diophantine equation 1=Σ1/ni + 1/Πni and a class of homologically trivial complex surface singularities", Pacific Journal of Mathematics 133 (1): 41–67

Brenton, Lawrence; Vasiliu, Ana (2002), "Znám's problem", Mathematics Magazine 75 (1): 3–11

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I don't suppose the decayed link is on the Internet Archive? –  Charles Oct 12 '11 at 15:33
    
@Charles It does not seem to be. –  deinst Oct 12 '11 at 15:43
    
Thank you! Looks like I got in way over my head ;). –  josh Oct 12 '11 at 21:41
    
@josh Don't let the first section of Brenton and Hill scare you, the rest is elementary number theory, and should be accessible to undergraduates. The missing $n=8$ solutions are the result of an undergraduate research project. –  deinst Oct 12 '11 at 23:54
    
@deinst I'll see how well I can handle it! –  josh Oct 13 '11 at 4:58
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As the link mentioned in the Mathematics Magazine article is inaccessible, I have rerun the code described in the article, and I report the results (122 octets for $n=8$) here. It appears that the run described in the article missed three of the octets. I assume that the missing octets are the ones mentioned in the OEIS article found by Zhao Hui Du.

2 3 7 43 1807 3263443  10650056950807 113423713055421844361000443
2 3 7 43 1807 3263443  10652778201539 41691378583707695
2 3 7 43 1807 3263443  10699597306267 2300171639909623
2 3 7 43 1807 3263443  10650057792155 134811739261383753719
2 3 7 43 1807 3263447  2130014000915 22684798220467498090185211
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 23316080984691959
2 3 7 43 1807 3263531  119666789791 8081907028348841339
2 3 7 43 1807 3263591  71480133827 761302020256877140089595
2 3 7 43 1807 3263779  31834629787 4396910340967
2 3 7 43 1807 3264187  14298637519 152316021000302785506427
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  637617223447 406555723635623909338363
2 3 7 43 1823 193667  637617223459 31273517203328870463055
2 3 7 43 1823 193675  4683210919 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 43 3263 4051  2558951 61088439723561979
2 3 7 43 3559 3667  33816127 797040720326433787
2 3 7 47 395 779731  607979652631 369639258012703445569531
2 3 7 47 395 779731  607979655287 139119028839856004123
2 3 7 47 395 779731  607979652683 6974325623477705424647
2 3 7 47 395 779731  607979793451 2624887933109395111
2 3 7 47 395 779731  607979652647 21743485766025360000683
2 3 7 47 395 779731  607979697799 8183472856913555659
2 3 7 47 395 779731  607979653531 410254449012081168631
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 779831  6020372531 3660733426607933569531
2 3 7 47 395 781727  305967719 125881309327
2 3 7 47 395 782111  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  15435513367 238255072887400163323
2 3 7 47 403 19403  15435513463 2456237880094942747
2 3 7 47 403 19403  15435513395 8215692183434294399
2 3 7 47 403 19403  15435516179 84697872837562655
2 3 7 47 415 8111  6644612311 44150872756848148411
2 3 7 47 415 8111  6644613463 38292177286592827
2 3 7 47 415 8111  6644612339 1522443894582665279
2 3 7 47 415 8111  6644645747 1320426321921983
2 3 7 47 449 4477  12137 34035763385
2 3 7 47 583 1223  1407479767 1980999293106894523
2 3 7 47 583 1223  2202310039 3899834875
2 3 7 47 583 1223  1468268915 33995520959
2 3 7 47 583 1223  1407479807 48317057302587443
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  10057317271 101149630679497570171
2 3 7 55 179 24323  10057317467 513449911932648503
2 3 7 55 179 24323  10057317311 2467064172726591731
2 3 7 55 179 24323  10057325347 12523178395739983
2 3 7 55 179 24323  10057317287 5949978284730273323
2 3 7 55 179 24323  10057320619 30202945461748519
2 3 7 55 179 24323  10057317967 145121431390804003
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 67 113 28925  48220169 4074021053
2 3 7 67 113 29153  3712777 45401353
2 3 7 67 113 34477  178945 178344158228021
2 3 7 67 187 283  334651 49836124516795
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 17 101 149  3109 52495396603
2 3 11 23 31 47059  2214502423 4904020979258368507
2 3 11 23 31 47059  2294166883 63772955407
2 3 11 23 31 47059  2215070383 8636647107907
2 3 11 23 31 47059  2214502831 11990273552017987
2 3 11 23 31 47059  2446798471 23325584587
2 3 11 23 31 47059  2214502475 92528699894575367
2 3 11 23 31 47059  2244604355 165128325167
2 3 11 23 31 47059  2214524099 226233749172527
2 3 11 23 31 47059  2214502427 980804197623275639
2 3 11 23 31 47059  2612824727 14526193019
2 3 11 23 31 47059  2217342227 1729101023519
2 3 11 23 31 47059  2214504467 2398056482005535
2 3 11 23 31 47059  3375982667 6436718855
2 3 11 23 31 47059  2214502687 18505741750517011
2 3 11 23 31 47059  2365012087 34797266971
2 3 11 23 31 47059  2214610807 45248521436443
2 3 11 23 31 47063  442938131 980970939025927675
2 3 11 23 31 47063  447473399 43702604167
2 3 11 23 31 47095  59897203 132743972247361531
2 3 11 23 31 47119  36349891 4619150372467
2 3 11 23 31 47131  30382063 67384091875543675
2 3 11 23 31 47147  24928579 11061526082145911
2 3 11 23 31 47243  12017087 26715920281613179
2 3 11 23 31 47423  6114059 13644326865136507
2 3 11 23 31 47479  5307047 2371471764522551
2 3 11 23 31 47491  5161279 4952592862147
2 3 11 23 31 49759  866923 2029951372029307
2 3 11 23 31 60563  211031 601432790177275
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 3 11 25 29 1097  2753 144508961851
2 3 11 31 35 67  369067 1770735487291
2 3 13 25 29 67  2981 11294561851
2 5 7 11 17 157  961 4398619
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