Problems & Puzzles: Puzzles

 Puzzle 1115 Follow up to Puzzle 1114... Let's now make a question related to the process described in Puzzle 1114 Here we ask for composite Mersenne integers such that the new integer obtained by the process used in the Puzzle 1114 is prime. For example: 2^11-1 is the composite 2047 = 23*89 but 2389 is prime. Bingo! Q. Send your largest composite Mersenne integer such that the concatenation in ascending order of its prime factor (without repetition) is prime.

During the week 17-23 December 2022, contributions came from Jean-Marc Rebert, Michael Branicky, Paul Cleary, Gennady Gusev, Emmanuel Vantieghem, Oscar Volpatti

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Jean-Marc wrote:

Composite Mersenne integer such that the concatenation in ascending order of its prime factor (without repetition) is prime.

2^6 - 1  = 63 = (3^2 * 7) is a 2-digit composite integer -> 37 is a 2-digit prime.
2^8 - 1  = 255 = (3 * 5 * 17) is a 3-digit composite integer -> 3517 is a 4-digit prime.
2^9 - 1  = 511 = (7 * 73) is a 3-digit composite integer -> 773 is a 3-digit prime.
2^11 - 1  = 2047 = (23 * 89) is a 4-digit composite integer -> 2389 is a 4-digit prime.
2^14 - 1  = 16383 = (3 * 43 * 127) is a 5-digit composite integer -> 343127 is a 6-digit prime.
2^32 - 1  = 4294967295 = (3 * 5 * 17 * 257 * 65537) is a 10-digit composite integer -> 351725765537 is a 12-digit prime.
2^36 - 1  = 68719476735 = (3^3 * 5 * 7 * 13 * 19 * 37 * 73 * 109) is a 11-digit composite integer -> 35713193773109 is a 14-digit prime.
2^40 - 1  = 1099511627775 = (3 * 5^2 * 11 * 17 * 31 * 41 * 61681) is a 13-digit composite integer -> 351117314161681 is a 15-digit prime.
2^48 - 1  = 281474976710655 = (3^2 * 5 * 7 * 13 * 17 * 97 * 241 * 257 * 673) is a 15-digit composite integer -> 357131797241257673 is a 18-digit prime.
2^66 - 1  = 73786976294838206463 = (3^2 * 7 * 23 * 67 * 89 * 683 * 20857 * 599479) is a 20-digit composite integer -> 3723678968320857599479 is a 22-digit prime.
2^75 - 1  = 37778931862957161709567 = (7 * 31 * 151 * 601 * 1801 * 100801 * 10567201) is a 23-digit composite integer -> 731151601180110080110567201 is a 27-digit prime.
2^79 - 1  = 604462909807314587353087 = (2687 * 202029703 * 1113491139767) is a 24-digit composite integer -> 26872020297031113491139767 is a 26-digit prime.
2^123 - 1  = 10633823966279326983230456482242756607 = (7 * 13367 * 3887047 * 164511353 * 177722253954175633) is a 38-digit composite integer -> 7133673887047164511353177722253954175633 is a 40-digit prime.
2^193 - 1  = 12554203470773361527671578846415332832204710888928069025791 = (13821503 * 61654440233248340616559 * 14732265321145317331353282383) is a 59-digit composite integer -> 138215036165444023324834061655914732265321145317331353282383 is a 60-digit prime.
2^222 - 1  = 6739986666787659948666753771754907668409286105635143120275902562303 = (3^2 * 7 * 223 * 1777 * 3331 * 17539 * 321679 * 25781083 * 26295457 * 319020217 * 616318177 * 107775231312019) is a 67-digit composite integer -> 3722317773331175393216792578108326295457319020217616318177107775231312019 is a 73-digit prime.
2^247 - 1  = 226156424291633194186662080095093570025917938800079226639565593765455331327 = (8191 * 15809 * 524287 * 6459570124697 * 402004106269663 * 1282816117617265060453496956212169) is a 75-digit composite integer -> 81911580952428764595701246974020041062696631282816117617265060453496956212169 is a 77-digit prime.
2^251 - 1  = 3618502788666131106986593281521497120414687020801267626233049500247285301247 = (503 * 54217 * 178230287214063289511 * 61676882198695257501367 * 12070396178249893039969681) is a 76-digit composite integer -> 503542171782302872140632895116167688219869525750136712070396178249893039969681 is a 78-digit prime.

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Michael wrote:

The largest I found was 2^3451-1, using complete factorizations at http://factordb.com

Other exponents found were 6, 8, 9, 11, 14, 32, 36, 40, 48, 66, 75, 79, 123, 193, 222, 247, 251, 850, 1152, 1440.

2^3451 - 1 = 127 * 233 * 239 * 1103 * 2089 * 20231 * 55217 * 131071 * 136417 * 3616649 * 10353001 * 121793911 *
62983048367 * 131105292137 * 29421156660560567325767 * 113480555808832720110908560531
75361113 *
970596583005459173652432922101
7810064201521004178349356202268282852670198911141357299732185324536769414538999508070197039 *
180101042928908193675050835906
343149211376923977468326957880026018186976761869245787335941494404681996959136881077325231403
993638021882799086906268009
8826049529941917734840088706743712329492782446861517724248804800061108697428514055236560054906024
78411907200576489969868
32194498545399402129769656402418902952034398562645808162861112630358438953681649510482399250717214980
1317279487266847135
503446102003793760074343310622757464221689160648272101428165984846304085400058085769033566859407595517845
529875732108363
9233535915736727824680837492729698395461056232525401409459656264310710873594877976848923543933744225150355665
00597549877
01235728977286570599129030848517973787400617560769314214319789410278083479942012426181437211053239076419754577362
7470288
806432946850618647202288935674329

and

1272332391103208920231552171310711364173616649103530011217939116298304836713110529213729421156660560567325767113480555808832
72011090856053175361113970
59658300545917365243292210178100642015210041783493562022682828526701989111413572997321853245367694
1453899950807019703918
010104292890819367505083590634314921137692397746832695788002601818697676186924578733594149440468199695
913688107732523140
399363802188279908690626800988260495299419177348400887067437123294927824468615177242488048000611086974285
140552365600549
0602478411907200576489969868321944985453994021297696564024189029520343985626458081628611126303584389536816495
10482399250
71721498013172794872668471355034461020037937600743433106227574642216891606482721014281659848463040854000580857690
3356685
940759551784552987573210836392335359157367278246808374927296983954610562325254014094596562643107108735948779768489235
439
3374422515035566500597549877012357289772865705991290308485179737874006175607693142143197894102780834799420124261814372110
53239076419754577362747028880
6432946850618647202288935674329 is prime (1051 digits), using isprime from sympy and also prime via Dario Alpern's site. Bingo!

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Paul wrote:

This is my largest composite Mersenne Integer I could find.

2 ^251 -1 is the composite 503542171782302872140632895116167688219869525750136712070396178249893039969681 is prime

I went up to the power of 334 with no further prime found.

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2^251-1 = 503*54217*178230287214063289511*61676882198695257501367* 12070396178249893039969681;

503542171782302872140632895116167688219869525750136712070396178249893039969681 is prime.

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Emmanuel wrote:

The exponents 79 and 193 give primes but my biggest result is :
2^251 - 1 = 503*54217*178230287214063289511*61676882198695257501367*12070396178249893039969681
and  503542171782302872140632895116167688219869525750136712070396178249893039969681  is prime.
I examined all further prime exponents <= 431, without success.

(out of the question : I also examined composite exponents. Those that give primes are 6, 8, 9, 14, 32, 36, 40, 48, 66, 75, 123, 222, 247  and no other <= 392).

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Oscar wrote:

The concatenation of all prime factors of a composite mersenne number M(n) produces a prime for the following exponents:
6, 8, 9, 11, 14, 32, 36, 40, 48, 66, 75, 79, 123, 193, 222, 247, 251, 850, 1152...
This sequence also contains exponents 1440 and 3451, but there may be further entries between 1152 and 3451.

After the given example n=11, next prime exponents are 79, 193, 251...
The attached file P1115ov.txt contains the 21 solutions I found so far.
For n=3451, the resulting number is a 1051-digit prime

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