Factorizations of concatenated prime numbers

This is a table of factorizations of “concatenated prime numbers”. These numbers are formed by taking the prime numbers from 2 up to some limit, and concatenating them all together into one number. The first few numbers of this type are: 2, 23, 235, 2357, 235711, 23571113, and 2357111317.

After 23571113, the numbers in the table are shown with dots representing one or more missing numbers. For example, 2357..17 is short for the number 2357111317, and 2357..29 is short for 2357111317192329.

The period character is used instead of the usual asterisk to mean “multiply”, since it makes the table easier to read.

The last number in a factorization is sometimes written as the letter “p” or “c” followed by a number. This notation means a prime or composite number, respectively, with the given number of digits. So, p85 means a prime number of 85 digits, and c104 means a composite number (whose factors are unknown) of 104 digits. The actual value of this last factor can be computed from the other factors.

In this table, the only “concatenated prime numbers” which are prime are 2, 23, and 2357. [See updates, below] Please let me know if you find any mistakes in this table, or have found any additional factors, or if you know the next prime number in this series.

First incomplete factorization: 2357..349
Last complete factorization: 2357..521


2 = PRIME
23 = PRIME
235 = 5.47
2357 = PRIME
235711 = 7.151.223
23571113 = 23.29.35339
2357..17 = 11.214282847
2357..19 = 7.4363.7717859
2357..23 = 61.478943.806801
2357..29 = 3.4243.185176472401
2357..31 = 17.283.1787.76753.357211
2357..37 = 7.67.67.151.4967701595369
2357..41 = 25391.889501.104364752351
2357..43 = 6899.164963.7515281.27558919
2357..47 = 1597.2801.5269410931806332951
2357..53 = 3.2311.1237278209.274784055330749
2357..59 = 17.906133.12846401.1191126125288819
2357..61 = 3.13.3390511326677.178258515898000387
2357..67 = 1019.2313161253378144566969023310693
2357..71 = 3.3.3.8730041915527145606449758346652473
2357..73 = 17.52733.26293517701186435480644832888393
2357..79 = 211.251.1489.29890227360205834316383307128051
2357..83 = 41378209.1476377774324699.3858432486690092813
2357..89 = 3.3.36263.10139578626034777.7122852423207105431971
2357..97 = 479.263334931.199319423543.93753283248830261744671
2357..101 = 3.3.3.144701.152203.10700354290243.370444734644515649265347
2357..103 = 262343078386137703.8984842793233304809156132914967801
2357..107 = 3.31.8226137.10103870671.304939320191990278692375683644213337
2357..109 = 107.23019967906561143281.956955188651271435198987426389824927
2357..113 = 3.3.3.7.13.281429.45599361955432273. p35
2357..127 = 643.10607.15661.4018837. p46
2357..131 = 3.1035752029104599.1047023254840727.88678579216845187. p19
2357..137 = 7.1861.6871.539447.2138419.34755519440888786692543. p27
2357..139 = 3.251.160231.70720609.6960652997.8369051281. p37
2357..149 = 13.13.17.59.14076659.370265039.36037426049. p44
2357..151 = 3.57738451.117748643366713517. p54
2357..157 = 107069.40722083.1995158689301. p57
2357..163 = 7.29.29599.737413.16533595423. p62
2357..167 = 14733071537.278229276707.6505153025537. p53
2357..173 = 3.3.31.67.71.79.1619.3919.699959123. p67
2357..179 = 61911427.33004801591308244093460638820179. p55
2357..181 = 3.3.193.6871829803981.984160102252973. p66
2357..191 = 13.97.167.1537690022857.226964259397663.87687770793038447519
            .160154056608794182723.2283732816082313986684152479
2357..193 = 3.11.29.557.1747.192926646339376423418933347. p68
2357..197 = 31.4816974392586326891872084844773520489505127. p62
2357..199 = 3.13.13.13.197.105613.4330517801
            .5502954190852198184124234451796585462038073. p45
2357..211 = 11.43.5483.733511.65892911291742926896163302121
            .3125568541217689315277975113751. p40
2357..223 = 59.61.787.330390951609644212919999. p85
2357..227 = 7.137.154313
            .146136279270895280678880836744405148990333772191301. p60 (Sam Chong)
2357..229 = 647. p118
2357..233 = 31.157.659. p117
2357..239 = 3.3.3.3.3.3.11.9084343.53404874448534840071487306149359. p84
2357..241 = 13.181.881.39081047667186446148508593173110063
            .50394127357598746888545910986174743077. p51 (Donovan Johnson)_
2357..251 = 3.7.17.71.26701.89136607103657
            .54290725767198980123376315097824574442986607719. p63 (SC)
2357..257 = 31517.472610915099. p120
2357..263 = 43.4071979211.1679788333087
            .132638207270881221255323402440482150228239. p74 (DJ)
2357..269 = 3.11.125693. p135
2357..271 = 19.46838908157
            .2461933169125535769713639083639611578977677394990241. p82 (SC)
2357..277 = 127.510430777.541295789.378444494508638079142367
            .28475493197611851339024962994788029096409041. p61
2357..281 = 11.83.253307.107783930539.103547886614462293
            .787665212805008428560409.89342055626849724105094049. p65 (DJ)
2357..283 = 606334586461.88577547579846682973920668362263
            .120084327060080362294548390666359
            .120844217778287124562149093554238437561. p40 (DJ)
2357..293 = 13.17.79.81395714265253088040956686461583
            .83393752462669798268233009801219995235992915589571746867579. p62 (SC)
2357..307 = 11.1627.966433279410457.108518474205215544112523
            .7102067243719626554227868240167393. p84 (DJ)
2357..311 = 23.101279.15436778485674892152416973119591. p125 (DJ)
2357..313 = 7.233.719.2017.25345487441. p146 (DJ)
2357..317 = 719496213832163304035156894278359085943. p130 (SC)
2357..331 = 7.71.24048071.188662178368355580143.1320859610392348153944861630769
            .216923389236198609558821678067350492090000009865599. p61 (SC)
2357..337 = 3.3863.10729183.54077980973
            .3015004170238185511004806931051461. p120 (DJ)
2357..347 = 89.2087.2109329.10014372271620700533397713773055971
            .1497562954541046535930805880040033899375103265867. p84 (SC)
2357..349 = 3.317.383.112692633407. c164
2357..353 = 7.163.1117.3881.6547223933. c164
2357..359 = 439. p184
2357..367 = 643.53549729.3897362220871782833539717852739. c149 (DJ)

Some of the following have not been verified (by me):

2357..373 = 3.3.3.3.4448013767.271749183659.2832411303185721987467
            .58902901917444574940765431. c123
2357..379 = 7.11.13.275453.25152903353. c177
2357..383 = 3.19.131.719.859.8741.19249.308857.315432743. c167
2357..389 = 109.509.709.797.5945575093.2021207429346699842213177. c157
2357..397 = 3.3.23.191. c200
2357..401 = 659.611633841004093.687018636860794953485021121527
            .1341363587794985571347403706577. p130
2357..409 = 3.3.7.59.59.181.241.596767.1205622079.870603945237227. c171
2357..419 = 232487.3282785807311. p196
2357..421 = 3.661.1125834225323. c202
2357..431 = 19.61.7741.701394433. p204
2357..433 = 3.13.101.1361. c216
2357..449 = 7.337.773.171317453.828589294108661206361047
            .10190641200355883477451413279077. p162
    *
    *
2357..521 = 17.3889. p260
Donovan Johnson (July 2003) reports that the following numbers are prime:
Number        Digits
***********   ******
2357..719     355
2357..1033    499
2357..2297    1171
2357..3037    1543 (probable prime)
2357..11927   5719 (probable prime)
He writes that he has tested all numbers from 2357..433 to 2357..100003 (9510 total numbers checked) and found no other (probable) primes. 2357..100003 is a 46540 digit number.
E-mail me at jrhowell@ix.netcom.com

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Last Updated: February 17, 2006.