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1 Attachment(s)
Learn to use "Attachments" button (see below).
Your posts with endless rows of meaningless numbers make readers' eyes bleed. |
[QUOTE=Batalov;616224]Learn to use "Attachments" button[/QUOTE]He's already demonstrated he knows how to use attachments: [URL]https://mersenneforum.org/showpost.php?p=616066&postcount=36[/URL]
[URL]https://mersenneforum.org/showpost.php?p=616041&postcount=25[/URL] He's choosing not to. Why, we can not say with certainty. Plain text is among the many supported attachment types: "Valid file extensions: 7z bz2 c cpp csv doc exe gif gz ini jpg log lz md5 npg odp ods odt pdf png ppt svg tgz [COLOR=Red][B]txt[/B][/COLOR] wav webp xls xz zip" I think that type allows up to 1MB per attachment. Zip allows 4MB IIRC. |
[QUOTE=Batalov;616224]Learn to use "Attachments" button (see below).
Your posts with endless rows of meaningless numbers make readers' eyes bleed.[/QUOTE] Noted, the reason to use code segments is to provide an option for fast scrolling in finding unfactored/untested LCS-generated prime exponents for a particular range. Only two more remaining data pieces of this kind are to be posted, for 26-bit (to list the LCS-generated prime exponents at the double-check (DC) wavefront and also show that all 26-bit prime exponents of Mersenne primes are among the LCS-generated prime exponents) and 29-bit LCS-generated prime exponents. Then the post content will resume its normal flow. |
1 Attachment(s)
While waiting for the remaining data on unfactored/untested LCS-generated prime exponents to be collected online,
let me post a histogram of the number of known Mersenne primes according to the base-2 length of their prime exponents from 2 bits to 27 bits. The Wolfram code used to plot the histogram is shown below. [code] SetDirectory[NotebookDirectory[]]; fname = NotebookDirectory[] <> "MpHistogram.jpg"; MpData = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933}; nMp = Length[MpData]; Mpl = ConstantArray[0, nMp]; base = 2; ic = 0; While[ic < nMp, ic++; Mpl[[ic]] = Length[IntegerDigits[MpData[[ic]], base]];]; Show[Histogram[Mpl, Mpl[[nMp]]]] Export[fname, Show[Histogram[Mpl, Mpl[[nMp]]]]] [/code] |
Out of [B][COLOR="Red"]12624[/COLOR][/B] [B][COLOR="Blue"]29[/COLOR][/B]-bit LCS-generated prime exponents, the Mersenne numbers of 7561 prime exponents are factored and the Mersenne numbers of 5063 prime exponents currently remain unfactored.
Said unfactored 29-bit prime exponents are listed below. [code] 268638569 268638647 268660963 268849379 268919069 268919207 268919423 268954601 269097749 269183977 269220629 269241883 269247773 269347049 269403307 269403487 269478889 269515273 269515319 269515327 269515349 269515391 269523733 269542181 269542319 269564327 269577193 269585543 269585597 269594851 269603603 269640473 269673241 269687573 269708053 269725867 269726161 269736169 269776511 269840611 269888221 269943577 269957353 269961449 269967881 269970409 269984537 270004381 270037219 270037459 270037483 270037489 270049919 270066457 270068027 270107417 270109397 270131993 270155933 270178697 270183283 270203501 270203669 270210209 270210389 270210517 270227071 270238489 270322453 270342377 270350569 270361369 270368057 270371239 270371333 270371351 270391529 270428317 270451789 270452033 270461929 270504163 272624617 272754793 272755111 272862847 273411049 274084067 274242737 274789601 274789607 274843421 274965553 274966229 275034427 275246057 275344361 275350297 275377129 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1 Attachment(s)
This last post with a list of LCS-generated prime exponents includes [B][COLOR="Red"]40254[/COLOR][/B] [B][COLOR="Blue"]26[/COLOR][/B]-bit prime exponents in the interval (2[SUP]25[/SUP] = 33554432, 2[SUP]26[/SUP] = 67108864).
Said exponents are generated from the LCS set in post #21, [url]https://mersenneforum.org/showpost.php?p=615845&postcount=21[/url], with the Wolfram code shown in post #25, [url]https://mersenneforum.org/showpost.php?p=616041&postcount=25[/url] (where [I]MnLength[/I] is set to 26 bits), and saved in a DAT file (see the attached ZIP file). Note that the total number of 26-bit prime exponents is 1894120. Among the generated prime exponents are two 26-bit prime exponents of Mersenne primes M[M]43112609[/M] and M[M]57885161[/M]. The two other 26-bit prime exponents of Mersenne primes, M[M]37156667[/M] and M[M]42643801[/M], are not listed as they can be generated only with alternative LCS sets involving [COLOR="Blue"]bit reversal[/COLOR] or [COLOR="blue"]cyclic folding[/COLOR] of the binary strings of the prime exponents. Out of the [B][COLOR="Red"]40254[/COLOR][/B] [B][COLOR="Blue"]26[/COLOR][/B]-bit LCS-generated prime exponents, [B]only [COLOR="Red"]450[/COLOR] [COLOR="Blue"]26[/COLOR]-bit LCS-generated prime exponents remain [COLOR="Blue"]unverified[/COLOR][/B]. Said unverified prime exponents are listed below. [code] 62016049 Unverified 62036137 Unverified 62269463 Unverified 62270179 Unverified 62270183 Unverified 62566627 Unverified 62585441 Unverified 62587673 Unverified 62753437 Unverified 62817049 Unverified 62930213 Unverified 62945513 Unverified 62945819 Unverified 62954653 Unverified 62954687 Unverified 62954779 Unverified 62954963 Unverified 62977037 Unverified 62977177 Unverified 62977181 Unverified 63016091 Unverified 63054617 Unverified 63058147 Unverified 63068221 Unverified 63068311 Unverified 63072413 Unverified 63075613 Unverified 63090389 Unverified 63090397 Unverified 63110869 Unverified 63116927 Unverified 63142621 Unverified 63156493 Unverified 63156617 Unverified 63156713 Unverified 63177497 Unverified 63191267 Unverified 63191269 Unverified 63204497 Unverified 63221957 Unverified 63226013 Unverified 63227341 Unverified 63230591 Unverified 63237169 Unverified 63277759 Unverified 63285931 Unverified 63290113 Unverified 63308569 Unverified 63328483 Unverified 63358147 Unverified 63358249 Unverified 63478549 Unverified 63494869 Unverified 63498679 Unverified 63574243 Unverified 63640939 Unverified 63647971 Unverified 63693593 Unverified 63721019 Unverified 63775513 Unverified 63802051 Unverified 63802087 Unverified 63959267 Unverified 63966493 Unverified 63970069 Unverified 63975889 Unverified 63983321 Unverified 63990833 Unverified 64012061 Unverified 64012573 Unverified 64013597 Unverified 64013951 Unverified 64016699 Unverified 64017001 Unverified 64019201 Unverified 64021457 Unverified 64024733 Unverified 64036109 Unverified 64037471 Unverified 64037747 Unverified 64045213 Unverified 64045759 Unverified 64093499 Unverified 64100593 Unverified 64102109 Unverified 64104041 Unverified 64104221 Unverified 64104659 Unverified 64106837 Unverified 64106929 Unverified 64109153 Unverified 64113349 Unverified 64113373 Unverified 64113383 Unverified 64113617 Unverified 64113713 Unverified 64113979 Unverified 64114103 Unverified 64114397 Unverified 64115327 Unverified 64116797 Unverified 64116821 Unverified 64116863 Unverified 64118399 Unverified 64118699 Unverified 64118701 Unverified 64118737 Unverified 64118981 Unverified 64119059 Unverified 64119467 Unverified 64119571 Unverified 64120613 Unverified 64121027 Unverified 64121173 Unverified 64122013 Unverified 64122239 Unverified 64124419 Unverified 64124429 Unverified 64124743 Unverified 64125653 Unverified 64125661 Unverified 64125671 Unverified 64125697 Unverified 64125833 Unverified 64126871 Unverified 64130741 Unverified 64130753 Unverified 64130797 Unverified 64131167 Unverified 64131269 Unverified 64131329 Unverified 64131341 Unverified 64133257 Unverified 64133273 Unverified 64133281 Unverified 64134527 Unverified 64134569 Unverified 64135277 Unverified 64135363 Unverified 64135399 Unverified 64135601 Unverified 64135901 Unverified 64136783 Unverified 64137851 Unverified 64137959 Unverified 64138157 Unverified 64139461 Unverified 64144669 Unverified 64144721 Unverified 64144853 Unverified 64145117 Unverified 64148129 Unverified 64148423 Unverified 64148893 Unverified 64148941 Unverified 64149457 Unverified 64151089 Unverified 64152371 Unverified 64153357 Unverified 64153363 Unverified 64153429 Unverified 64153513 Unverified 64153757 Unverified 64153951 Unverified 64153993 Unverified 64154693 Unverified 64155697 Unverified 64156471 Unverified 64157329 Unverified 64158769 Unverified 64160167 Unverified 64160177 Unverified 64162961 Unverified 64163383 Unverified 64163699 Unverified 64164739 Unverified 64164781 Unverified 64165121 Unverified 64166741 Unverified 64167349 Unverified 64167377 Unverified 64169389 Unverified 64169449 Unverified 64171741 Unverified 64172203 Unverified 64172341 Unverified 64172377 Unverified 64172497 Unverified 64172639 Unverified 64172741 Unverified 64173349 Unverified 64173371 Unverified 64173883 Unverified 64174057 Unverified 64174769 Unverified 64174783 Unverified 64174897 Unverified 64175081 Unverified 64176059 Unverified 64176089 Unverified 64176097 Unverified 64176113 Unverified 64176251 Unverified 64178141 Unverified 64178291 Unverified 64178347 Unverified 64178353 Unverified 64179209 Unverified 64179229 Unverified 64181197 Unverified 64181951 Unverified 64182221 Unverified 64183913 Unverified 64184551 Unverified 64184671 Unverified 64184713 Unverified 64184909 Unverified 64184969 Unverified 64184987 Unverified 64185089 Unverified 64187077 Unverified 64187107 Unverified 64187173 Unverified 64187369 Unverified 64188647 Unverified 64189409 Unverified 64190513 Unverified 64194199 Unverified 64194239 Unverified 64194931 Unverified 64194953 Unverified 64198889 Unverified 64201387 Unverified 64201933 Unverified 64202189 Unverified 64202401 Unverified 64203409 Unverified 64204909 Unverified 64204991 Unverified 64205081 Unverified 64205137 Unverified 64205171 Unverified 64206139 Unverified 64207259 Unverified 64207301 Unverified 64207849 Unverified 64212073 Unverified 64212671 Unverified 64213343 Unverified 64213819 Unverified 64213993 Unverified 64215163 Unverified 64215829 Unverified 64217137 Unverified 64219451 Unverified 64219499 Unverified 64303153 Unverified 64366409 Unverified 64366439 Unverified 64446749 Unverified 64447147 Unverified 64447189 Unverified 64475743 Unverified 64527469 Unverified 64528489 Unverified 64529177 Unverified 64547281 Unverified 64618111 Unverified 64643357 Unverified 64643387 Unverified 64649449 Unverified 64657553 Unverified 64689479 Unverified 64689563 Unverified 64689629 Unverified 64696547 Unverified 64714151 Unverified 64729403 Unverified 64774937 Unverified 64803043 Unverified 64867967 Unverified 64931147 Unverified 64931179 Unverified 64979737 Unverified 65006569 Unverified 65027357 Unverified 65027377 Unverified 65037151 Unverified 65088581 Unverified 65092519 Unverified 65128093 Unverified 65128681 Unverified 65132671 Unverified 65165543 Unverified 65173153 Unverified 65173457 Unverified 65187131 Unverified 65212031 Unverified 65214827 Unverified 65228393 Unverified 65228563 Unverified 65228777 Unverified 65250589 Unverified 65250727 Unverified 65253841 Unverified 65260621 Unverified 65260721 Unverified 65294497 Unverified 65301581 Unverified 65312627 Unverified 65313557 Unverified 65319187 Unverified 65324141 Unverified 65324317 Unverified 65327987 Unverified 65351857 Unverified 65351911 Unverified 65352031 Unverified 65353961 Unverified 65374873 Unverified 65374927 Unverified 65374937 Unverified 65382557 Unverified 65414941 Unverified 65414959 Unverified 65431867 Unverified 65445179 Unverified 65455463 Unverified 65476841 Unverified 65487641 Unverified 65495657 Unverified 65509799 Unverified 65565079 Unverified 65595823 Unverified 65610521 Unverified 65778397 Unverified 65778547 Unverified 65818171 Unverified 65889049 Unverified 65899033 Unverified 65899139 Unverified 65899159 Unverified 65899189 Unverified 65899219 Unverified 65899241 Unverified 65916637 Unverified 65916787 Unverified 65966237 Unverified 65979569 Unverified 65979689 Unverified 65979707 Unverified 65979751 Unverified 66091573 Unverified 66091589 Unverified 66111197 Unverified 66140209 Unverified 66140777 Unverified 66140849 Unverified 66153449 Unverified 66171107 Unverified 66179977 Unverified 66180007 Unverified 66180073 Unverified 66263273 Unverified 66429721 Unverified 66450649 Unverified 66463561 Unverified 66463849 Unverified 66463871 Unverified 66464059 Unverified 66464417 Unverified 66464483 Unverified 66498787 Unverified 66543901 Unverified 66544301 Unverified 66558677 Unverified 66579433 Unverified 66600223 Unverified 66600229 Unverified 66609821 Unverified 66611357 Unverified 66613693 Unverified 66613709 Unverified 66624739 Unverified 66624839 Unverified 66640661 Unverified 66644377 Unverified 66644429 Unverified 66682171 Unverified 66684977 Unverified 66689813 Unverified 66700057 Unverified 66705563 Unverified 66705629 Unverified 66711779 Unverified 66715349 Unverified 66715507 Unverified 66730141 Unverified 66733781 Unverified 66743273 Unverified 66746083 Unverified 66746087 Unverified 66746249 Unverified 66779753 Unverified 66779759 Unverified 66780209 Unverified 66784021 Unverified 66786229 Unverified 66786347 Unverified 66786359 Unverified 66786563 Unverified 66786781 Unverified 66786793 Unverified 66801409 Unverified 66817447 Unverified 66826499 Unverified 66847513 Unverified 66848477 Unverified 66866599 Unverified 66867277 Unverified 66874513 Unverified 66874559 Unverified 66876569 Unverified 66876643 Unverified 66897211 Unverified 66912137 Unverified 66924829 Unverified 66924967 Unverified 66926749 Unverified 66933931 Unverified 66944299 Unverified 66947707 Unverified 66947761 Unverified 66947801 Unverified 66947827 Unverified 66956069 Unverified 66956237 Unverified 66960017 Unverified 66960067 Unverified 66964529 Unverified 66996647 Unverified 67015381 Unverified 67026587 Unverified 67028113 Unverified 67028239 Unverified 67028243 Unverified 67036817 Unverified 67036867 Unverified 67061993 Unverified 67068569 Unverified[/code] |
All prime exponents of known Mersenne primes can give at least one [B][COLOR="Red"]prime number[/COLOR][/B] after [B][COLOR="Blue"]bit reversal[/COLOR][/B], [B][COLOR="blue"]base-2 cyclic rotation[/COLOR][/B], or [B][COLOR="blue"]base-2 cyclic rotation after bit reversal[/COLOR][/B] [COLOR="Red"][B]except[/B][/COLOR] [COLOR="Blue"]M[M]21701[/M][/COLOR] and [COLOR="Blue"]M[M]756839[/M][/COLOR].
[code] #1, Prime Exponent 2 2(Cyclic Rotation 1 Position to the Right after Bit Reversal) --------- #2, Prime Exponent 3 3(Bit Reversal) 3(Cyclic Rotation 1 Position to the Right) 3(Cyclic Rotation 1 Position to the Right after Bit Reversal) --------- #3, Prime Exponent 5 5(Bit Reversal) 3(Cyclic Rotation 2 Positions to the Right) 3(Cyclic Rotation 2 Positions to the Right after Bit Reversal) --------- #4, Prime Exponent 7 7(Bit Reversal) 7(Cyclic Rotation 1 Position to the Right) 7(Cyclic Rotation 2 Positions to the Right) 7(Cyclic Rotation 1 Position to the Right after Bit Reversal) 7(Cyclic Rotation 2 Positions to the Right after Bit Reversal) --------- #5, Prime Exponent 13 11(Bit Reversal) 7(Cyclic Rotation 2 Positions to the Right) 11(Cyclic Rotation 3 Positions to the Right) 13(Cyclic Rotation 1 Position to the Right after Bit Reversal) 7(Cyclic Rotation 3 Positions to the Right after Bit Reversal) --------- #6, Prime Exponent 17 17(Bit Reversal) 3(Cyclic Rotation 4 Positions to the Right) 3(Cyclic Rotation 4 Positions to the Right after Bit Reversal) --------- #7, Prime Exponent 19 7(Cyclic Rotation 4 Positions to the Right) 7(Cyclic Rotation 3 Positions to the Right after Bit Reversal) 19(Cyclic Rotation 4 Positions to the Right after Bit Reversal) --------- #8, Prime Exponent 31 31(Bit Reversal) 31(Cyclic Rotation 1 Position to the Right) 31(Cyclic Rotation 2 Positions to the Right) 31(Cyclic Rotation 3 Positions to the Right) 31(Cyclic Rotation 4 Positions to the Right) 31(Cyclic Rotation 1 Position to the Right after Bit Reversal) 31(Cyclic Rotation 2 Positions to the Right after Bit Reversal) 31(Cyclic Rotation 3 Positions to the Right after Bit Reversal) 31(Cyclic Rotation 4 Positions to the Right after Bit Reversal) --------- #9, Prime Exponent 61 47(Bit Reversal) 31(Cyclic Rotation 2 Positions to the Right) 47(Cyclic Rotation 3 Positions to the Right) 59(Cyclic Rotation 5 Positions to the Right) 59(Cyclic Rotation 2 Positions to the Right after Bit Reversal) 61(Cyclic Rotation 3 Positions to the Right after Bit Reversal) 31(Cyclic Rotation 5 Positions to the Right after Bit Reversal) --------- #10, Prime Exponent 89 89(Cyclic Rotation 3 Positions to the Right after Bit Reversal) --------- #11, Prime Exponent 107 107(Bit Reversal) 61(Cyclic Rotation 3 Positions to the Right) 47(Cyclic Rotation 5 Positions to the Right) 61(Cyclic Rotation 3 Positions to the Right after Bit Reversal) 47(Cyclic Rotation 5 Positions to the Right after Bit Reversal) --------- #12, Prime Exponent 127 127(Bit Reversal) 127(Cyclic Rotation 1 Position to the Right) 127(Cyclic Rotation 2 Positions to the Right) 127(Cyclic Rotation 3 Positions to the Right) 127(Cyclic Rotation 4 Positions to the Right) 127(Cyclic Rotation 5 Positions to the Right) 127(Cyclic Rotation 6 Positions to the Right) 127(Cyclic Rotation 1 Position to the Right after Bit Reversal) 127(Cyclic Rotation 2 Positions to the Right after Bit Reversal) 127(Cyclic Rotation 3 Positions to the Right after Bit Reversal) 127(Cyclic Rotation 4 Positions to the Right after Bit Reversal) 127(Cyclic Rotation 5 Positions to the Right after Bit Reversal) 127(Cyclic Rotation 6 Positions to the Right after Bit Reversal) --------- #13, Prime Exponent 521 577(Bit Reversal) 193(Cyclic Rotation 3 Positions to the Right) 19(Cyclic Rotation 9 Positions to the Right) 131(Cyclic Rotation 9 Positions to the Right after Bit Reversal) --------- #14, Prime Exponent 607 919(Cyclic Rotation 2 Positions to the Right) 971(Cyclic Rotation 3 Positions to the Right) 997(Cyclic Rotation 4 Positions to the Right) 191(Cyclic Rotation 9 Positions to the Right) --------- #15, Prime Exponent 1279 1663(Cyclic Rotation 1 Position to the Right) 1951(Cyclic Rotation 3 Positions to the Right) 1999(Cyclic Rotation 4 Positions to the Right) 1279(Cyclic Rotation 4 Positions to the Right after Bit Reversal) 1663(Cyclic Rotation 5 Positions to the Right after Bit Reversal) 1951(Cyclic Rotation 7 Positions to the Right after Bit Reversal) 1999(Cyclic Rotation 8 Positions to the Right after Bit Reversal) --------- #16, Prime Exponent 2203 1811(Cyclic Rotation 3 Positions to the Right) 2953(Cyclic Rotation 4 Positions to the Right) 881(Cyclic Rotation 7 Positions to the Right) 311(Cyclic Rotation 11 Positions to the Right) 571(Cyclic Rotation 7 Positions to the Right after Bit Reversal) 2333(Cyclic Rotation 8 Positions to the Right after Bit Reversal) 1607(Cyclic Rotation 10 Positions to the Right after Bit Reversal) 2851(Cyclic Rotation 11 Positions to the Right after Bit Reversal) --------- #17, Prime Exponent 2281 2417(Bit Reversal) 797(Cyclic Rotation 3 Positions to the Right) 1223(Cyclic Rotation 5 Positions to the Right) 2659(Cyclic Rotation 6 Positions to the Right) 467(Cyclic Rotation 11 Positions to the Right) 2251(Cyclic Rotation 5 Positions to the Right after Bit Reversal) 739(Cyclic Rotation 11 Positions to the Right after Bit Reversal) --------- #18, Prime Exponent 3217 457(Cyclic Rotation 4 Positions to the Right) 569(Cyclic Rotation 7 Positions to the Right) 2339(Cyclic Rotation 11 Positions to the Right) --------- #19, Prime Exponent 4253 5651(Cyclic Rotation 3 Positions to the Right) 1889(Cyclic Rotation 7 Positions to the Right) --------- #20, Prime Exponent 4423 4423(Cyclic Rotation 11 Positions to the Right after Bit Reversal) --------- #21, Prime Exponent 9689 3259(Cyclic Rotation 3 Positions to the Right) 6551(Cyclic Rotation 6 Positions to the Right) 11467(Cyclic Rotation 7 Positions to the Right) 7577(Cyclic Rotation 10 Positions to the Right) 4919(Cyclic Rotation 5 Positions to the Right after Bit Reversal) 10651(Cyclic Rotation 6 Positions to the Right after Bit Reversal) 3539(Cyclic Rotation 13 Positions to the Right after Bit Reversal) --------- #22, Prime Exponent 9941 6581(Cyclic Rotation 2 Positions to the Right) 5741(Cyclic Rotation 4 Positions to the Right) 5531(Cyclic Rotation 6 Positions to the Right) 10957(Cyclic Rotation 7 Positions to the Right) 3499(Cyclic Rotation 13 Positions to the Right) 9901(Cyclic Rotation 4 Positions to the Right after Bit Reversal) 6571(Cyclic Rotation 6 Positions to the Right after Bit Reversal) --------- #23, Prime Exponent 11213 6899(Cyclic Rotation 2 Positions to the Right) 13163(Cyclic Rotation 8 Positions to the Right) 7789(Cyclic Rotation 11 Positions to the Right) 6043(Cyclic Rotation 13 Positions to the Right) 5839(Cyclic Rotation 4 Positions to the Right after Bit Reversal) 3931(Cyclic Rotation 10 Positions to the Right after Bit Reversal) --------- #24, Prime Exponent 19937 17207(Cyclic Rotation 6 Positions to the Right) --------- #25, Prime Exponent 21701 None --------- #26, Prime Exponent 23209 6997(Cyclic Rotation 3 Positions to the Right) 9941(Cyclic Rotation 5 Positions to the Right) 10861(Cyclic Rotation 9 Positions to the Right) --------- #27, Prime Exponent 44497 29803(Cyclic Rotation 10 Positions to the Right) 23459(Cyclic Rotation 15 Positions to the Right) --------- #28, Prime Exponent 86243 14983(Cyclic Rotation 5 Positions to the Right) 34589(Cyclic Rotation 14 Positions to the Right) 14423(Cyclic Rotation 15 Positions to the Right after Bit Reversal) --------- #29, Prime Exponent 110503 125929(Cyclic Rotation 2 Positions to the Right) 108503(Cyclic Rotation 9 Positions to the Right) 125429(Cyclic Rotation 11 Positions to the Right) 48799(Cyclic Rotation 15 Positions to the Right) 120779(Cyclic Rotation 8 Positions to the Right after Bit Reversal) 77743(Cyclic Rotation 15 Positions to the Right after Bit Reversal) --------- #30, Prime Exponent 132049 166919(Cyclic Rotation 7 Positions to the Right) 214531(Cyclic Rotation 8 Positions to the Right) 1583(Cyclic Rotation 8 Positions to the Right after Bit Reversal) 197003(Cyclic Rotation 10 Positions to the Right after Bit Reversal) 122929(Cyclic Rotation 13 Positions to the Right after Bit Reversal) --------- #31, Prime Exponent 216091 49597(Cyclic Rotation 14 Positions to the Right) 134333(Cyclic Rotation 14 Positions to the Right after Bit Reversal) 99119(Cyclic Rotation 16 Positions to the Right after Bit Reversal) --------- #32, Prime Exponent 756839 None --------- #33, Prime Exponent 859433 609163(Bit Reversal) 171293(Cyclic Rotation 8 Positions to the Right) 676771(Cyclic Rotation 11 Positions to the Right) 810149(Cyclic Rotation 9 Positions to the Right after Bit Reversal) 309433(Cyclic Rotation 16 Positions to the Right after Bit Reversal) 169751(Cyclic Rotation 19 Positions to the Right after Bit Reversal) --------- #34, Prime Exponent 1257787 418423(Cyclic Rotation 20 Positions to the Right) --------- #35, Prime Exponent 1398269 1485503(Cyclic Rotation 3 Positions to the Right) 2009783(Cyclic Rotation 16 Positions to the Right after Bit Reversal) --------- #36, Prime Exponent 2976221 3073453(Bit Reversal) 978349(Cyclic Rotation 11 Positions to the Right) 1758139(Cyclic Rotation 21 Positions to the Right) --------- #37, Prime Exponent 3021377 2888387(Cyclic Rotation 3 Positions to the Right after Bit Reversal) 241739(Cyclic Rotation 9 Positions to the Right after Bit Reversal) 1603081(Cyclic Rotation 12 Positions to the Right after Bit Reversal) --------- #38, Prime Exponent 6972593 2509597(Cyclic Rotation 19 Positions to the Right) 2724551(Cyclic Rotation 21 Positions to the Right) 3028121(Cyclic Rotation 5 Positions to the Right after Bit Reversal) --------- #39, Prime Exponent 13466917 16029493(Cyclic Rotation 14 Positions to the Right) 11510969(Cyclic Rotation 19 Positions to the Right) --------- #40, Prime Exponent 20996011 27275221(Cyclic Rotation 1 Position to the Right) 22495327(Cyclic Rotation 8 Positions to the Right) 32172043(Cyclic Rotation 11 Positions to the Right) 16604417(Cyclic Rotation 14 Positions to the Right) --------- #41, Prime Exponent 24036583 28810381(Cyclic Rotation 7 Positions to the Right after Bit Reversal) 3632953(Cyclic Rotation 14 Positions to the Right after Bit Reversal) 26327929(Cyclic Rotation 19 Positions to the Right after Bit Reversal) --------- #42, Prime Exponent 25964951 1625059(Cyclic Rotation 18 Positions to the Right) 3196511(Cyclic Rotation 23 Positions to the Right) 4102339(Cyclic Rotation 5 Positions to the Right after Bit Reversal) --------- #43, Prime Exponent 30402457 20774521(Cyclic Rotation 4 Positions to the Right) 30402457(Cyclic Rotation 17 Positions to the Right after Bit Reversal) 20774521(Cyclic Rotation 21 Positions to the Right after Bit Reversal) --------- #44, Prime Exponent 32582657 4710413(Cyclic Rotation 11 Positions to the Right after Bit Reversal) 21560321(Cyclic Rotation 14 Positions to the Right after Bit Reversal) --------- #45, Prime Exponent 37156667 57916849(Bit Reversal) 7204471(Cyclic Rotation 25 Positions to the Right) 49727347(Cyclic Rotation 16 Positions to the Right after Bit Reversal) 41399867(Cyclic Rotation 21 Positions to the Right after Bit Reversal) --------- #46, Prime Exponent 42643801 40413973(Cyclic Rotation 4 Positions to the Right) 5606093(Cyclic Rotation 23 Positions to the Right) 22111907(Cyclic Rotation 12 Positions to the Right after Bit Reversal) 21407923(Cyclic Rotation 21 Positions to the Right after Bit Reversal) --------- #47, Prime Exponent 43112609 34893349(Bit Reversal) 5296699(Cyclic Rotation 11 Positions to the Right) 36013339(Cyclic Rotation 10 Positions to the Right after Bit Reversal) 48388177(Cyclic Rotation 14 Positions to the Right after Bit Reversal) --------- #48, Prime Exponent 57885161 15624253(Cyclic Rotation 3 Positions to the Right) 42193849(Cyclic Rotation 17 Positions to the Right) 53413277(Cyclic Rotation 1 Position to the Right after Bit Reversal) 48619699(Cyclic Rotation 4 Positions to the Right after Bit Reversal) 15621643(Cyclic Rotation 8 Positions to the Right after Bit Reversal) 34290583(Cyclic Rotation 18 Positions to the Right after Bit Reversal) 12326519(Cyclic Rotation 25 Positions to the Right after Bit Reversal) --------- #49, Prime Exponent 74207281 73421233(Bit Reversal) 72364429(Cyclic Rotation 19 Positions to the Right) --------- #50, Prime Exponent 77232917 11311739(Cyclic Rotation 8 Positions to the Right) 41620883(Cyclic Rotation 19 Positions to the Right) --------- #51, Prime Exponent 82589933 70839709(Cyclic Rotation 19 Positions to the Right) 97368289(Cyclic Rotation 8 Positions to the Right after Bit Reversal) 102472561(Cyclic Rotation 15 Positions to the Right after Bit Reversal) ---------[/code] [code] (* Wolfram code *) MpData = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933}; nMp = Length[MpData]; base = 2; ic = 0; While[ic < nMp, ic++; count = 0; Print["#", ic, ", Prime Exponent ", MpData[[ic]]]; intlen1 = Length[IntegerDigits[MpData[[ic]], base]]; s1 = IntegerDigits[MpData[[ic]], base, intlen1]; s1r = IntegerDigits[IntegerReverse[MpData[[ic]], base], base, intlen1]; Mpi = IntegerReverse[MpData[[ic]], base]; If[(PrimeQ[Mpi] == True), count++; Print[Mpi, "(Bit Reversal)"];]; jc = 0; While[jc < intlen1 - 1, jc++; s1 = RotateRight[s1]; s1i = FromDigits[s1, base]; If[(PrimeQ[s1i] == True), count++; pl = " "; If[jc > 1, pl = "s ";]; Print[s1i, "(Cyclic Rotation ", jc, " Position", pl, "to the Right)"];];]; jc = 0; While[jc < intlen1 - 1, jc++; s1r = RotateRight[s1r]; s1ri = FromDigits[s1r, base]; If[(PrimeQ[s1ri] == True), count++; pl = " "; If[jc > 1, pl = "s ";]; Print[s1ri, "(Cyclic Rotation ", jc, " Position", pl, "to the Right after Bit Reversal)"];];]; If[count == 0, Print["None"];]; Print["---------"];];[/code] |
[QUOTE=Dobri;616318]All prime exponents of known Mersenne primes can give at least one [B][COLOR=Red]prime number[/COLOR][/B] after [B][COLOR=Blue]bit reversal[/COLOR][/B], [B][COLOR=blue]base-2 cyclic rotation[/COLOR][/B], or [B][COLOR=blue]base-2 cyclic rotation after bit reversal[/COLOR][/B] [COLOR=Red][B]except[/B][/COLOR] [COLOR=Blue]M[M]21701[/M][/COLOR] and [COLOR=Blue]M[M]756839[/M][/COLOR].
[/QUOTE]And if I scanned the list correctly, all the Mersenne numbers those primes correspond to, are either already-known Mersenne primes, factored Mersennes, or proven composites by DC or PRP/proof/cert, except for two, which have single LL composite residues, and are currently in progress in LLDC: M[M]73421233[/M] M[M]72364429[/M] In no case did these manipulations lead to a prime exponent of a known Mersenne prime with exponent greater than the starting point, or to an untested unfactored Mersenne number's exponent. In five cases, they lead only to the same exponent. ( 2 3 7 31 127 ) |
The [B][COLOR="Red"]mean[/COLOR][/B] and the [B][COLOR="Blue"]standard deviation[/COLOR][/B] of the [B][COLOR="Red"]normalized lengths[/COLOR][/B] of the longest common contiguous subsequences (LCSs) for [U]all pairs ([I]i[/I], [I]j[/I]), [I]i[/I] < [I]j[/I], for [I]i[/I], [I]j[/I] = 1, 2,..., 51, of the exponents of known Mersenne primes[/U] are as follows:
[B][COLOR="Red"]Mean[/COLOR][/B] ± [B][COLOR="Blue"]S.D.[/COLOR][/B] ≈ [B][COLOR="Red"]0.345[/COLOR][/B] ± [B][COLOR="Blue"]0.111[/COLOR][/B]. [code] (* Wolfram code *) MpData = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933}; nMp = Length[MpData]; lcsl = ConstantArray[0, nMp*(nMp - 1)/2]; base = 2; count = 0; ic = 0; While[ic < nMp, ic++; jc = 0; While[jc < nMp, jc++; If[ic < jc, count++; intlen1 = Length[IntegerDigits[MpData[[ic]], base]]; intlen2 = Length[IntegerDigits[MpData[[jc]], base]]; lcsl[[count]] = Length[LongestCommonSubsequence[IntegerDigits[MpData[[ic]], base, intlen1], IntegerDigits[MpData[[jc]], base, intlen2]]]/((intlen1 + intlen2)/2);];];]; lcsmean = N[Mean[lcsl]]; lcssd = N[StandardDeviation[lcsl]]; Print["Mean ± S.D. = ", lcsmean, " ± ", lcssd]; [/code] The [B][COLOR="Red"]mean[/COLOR][/B] and the [B][COLOR="Blue"]standard deviation[/COLOR][/B] of the [B][COLOR="Red"]normalized lengths[/COLOR][/B] of the LCSs for [U]arbitrary pairs of prime numbers[/U] (in the same range as the prime exponents of Mersenne primes) are as follows: [B][COLOR="Red"]Mean[/COLOR][/B] ± [B][COLOR="Blue"]S.D.[/COLOR][/B] ≈ [B][COLOR="Red"]0.309[/COLOR][/B] ± [B][COLOR="Blue"]0.064[/COLOR][/B]. [code] (* Wolfram code *) np = 1000; prange = PrimePi[82589933]; lcsl = ConstantArray[0, np*np]; base = 2; count = 0; ic = 0; While[ic < np, ic++; jc = 0; While[jc < np, jc++; count++; pic = Prime[RandomInteger[{1, prange}]]; intlen1 = Length[IntegerDigits[pic, base]]; pjc = pic; While[pjc == pic, pjc = Prime[RandomInteger[{1, prange}]];]; intlen2 = Length[IntegerDigits[pjc, base]]; lcsl[[count]] = Length[LongestCommonSubsequence[IntegerDigits[pic, base, intlen1], IntegerDigits[pjc, base, intlen2]]]/((intlen1 + intlen2)/2); ];]; lcsmean = N[Mean[lcsl]]; lcssd = N[StandardDeviation[lcsl]]; Print["Mean ± S.D. = ", lcsmean, " ± ", lcssd]; [/code] |
The [B][COLOR="Red"]mean[/COLOR][/B] and the [B][COLOR="Blue"]standard deviation[/COLOR][/B] of the [B][COLOR="Red"]normalized lengths[/COLOR][/B] of the longest common contiguous subsequences (LCSs) for [U]the 50 pairs of consecutive prime exponents of known Mersenne primes[/U] are as follows:
[B][COLOR="Red"]Mean[/COLOR][/B] ± [B][COLOR="Blue"]S.D.[/COLOR][/B] ≈ [B][COLOR="Red"]0.385[/COLOR][/B] ± [B][COLOR="Blue"]0.119[/COLOR][/B]. [code] (* Wolfram code *) MpData = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933}; nMp = Length[MpData]; lcsl = ConstantArray[0, nMp - 1]; base = 2; count = 0; ic = 1; While[ic < nMp, ic++; count++; intlen1 = Length[IntegerDigits[MpData[[ic]], base]]; intlen2 = Length[IntegerDigits[MpData[[ic - 1]], base]]; lcsl[[count]] = Length[LongestCommonSubsequence[IntegerDigits[MpData[[ic]], base, intlen1], IntegerDigits[MpData[[ic - 1]], base, intlen2]]]/((intlen1 + intlen2)/2);]; lcsmean = N[Mean[lcsl]]; lcssd = N[StandardDeviation[lcsl]]; Print["Mean ± S.D. = ", lcsmean, " ± ", lcssd]; [/code] |
1 Attachment(s)
The attached image shows the current [COLOR=Red][B]downward[/B][/COLOR] tendency of the linear fit of the [COLOR=Blue][B]normalized lengths[/B][/COLOR] of the longest common contiguous subsequences (LCSs) for [U]the 50 pairs of consecutive exponents of known Mersenne primes[/U].
For the graph of non-normalized lenghts, see post #8, [URL]https://mersenneforum.org/showpost.php?p=615342&postcount=8[/URL]. [code] (*Wolfram code*) SetDirectory[NotebookDirectory[]]; fname = NotebookDirectory[] <> "LCSnormalized.jpg"; MpData = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933}; nMp = Length[MpData]; lcsl = ConstantArray[0, nMp - 1]; base = 2; count = 0; ic = 1; While[ic < nMp, ic++; count++; intlen1 = Length[IntegerDigits[MpData[[ic]], base]]; intlen2 = Length[IntegerDigits[MpData[[ic - 1]], base]]; lcsl[[count]] = Length[LongestCommonSubsequence[IntegerDigits[MpData[[ic]], base, intlen1], IntegerDigits[MpData[[ic - 1]], base, intlen2]]]/((intlen1 + intlen2)/2);]; lcslfit = LinearModelFit[lcsl, x, x]; Show[BarChart[lcsl], Plot[lcslfit[x], {x, 1, nMp}], Frame -> True] Export[fname, Show[BarChart[lcsl], Plot[lcslfit[x], {x, 1, nMp}], Frame -> True]][/code] |
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