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 2020-12-24, 18:00 #3 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 2×5×773 Posts Straights These are exponents such as 160456789 which also sometimes come up in dubious claims or guesses as exponents of Mersenne primes. In the mersenne.org search space 2
 2020-12-24, 18:07 #4 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 2×5×773 Posts Repdigits and near-repdigits Repdigits are numbers consisting only of repetition of the same digit value. (Single-digit numbers are excluded by definition.) For Mersennes' exponents, repdigit digit values larger than one lead to composite exponents, and thereby to composite Mersenne numbers. Composite digit counts also lead to composite exponents and composite Mersenne numbers. This leaves exponents consisting of a prime number of ones, which lead to mostly composite exponents. It also leads to a very small set of candidates, since for p<109, number of digits no greater than 9, only 2, 3, 5, or 7 digits are prime length for exponent expressed in base ten. In 10, base 10 use ntheory; $count=0; for ($i=1; $i<10;$i++ ) { #repfield is $i as digits x 8 places foreach$j ( 1, 3, 7, 9 ) { #rightmost if ($i !=$j ) { $k=$i*11111111*10+$j; if ( Math::Prime::Util::GMP::is_prime($k) == 0 ) { # print "$k is composite\n"; } else { print "$k\n"; $count++; } } } } print "Counted$count\n(end)\n"; Requiring the differing digit to be on the right is a special case / subset of the near-repdigit definition. Checking for other positions of the differing digit, and annotating the resulting output with current status: Code: # nearrep2.pl # perl script to find near repdigit prime exponents i..iji..i, j != i, i>0, base 10 # where leftmost digit may be j but rightmost is not use ntheory; $count=0; for ($l=1; $l<9;$l++ ) { #power of ten at which digit differs for ( $j=0;$j<10; $j++ ) { #differing digit foreach$i ( 1, 3, 7, 9 ) { #repfield is $i as digits x 8 places if ($i != $j ) {$k= $i*111111111 +($j- $i) *10**$l; if ( Math::Prime::Util::GMP::is_prime($k) == 0 ) { # print "$k is composite\n"; } else { print "$k\n";$count++; } } } } } print "Counted $count\n(end)\n"; Code: 333333313 factored 999999929 NF 86; P-1 NF; PRP C, CERT 111111131 factored 111111181 LL C; LLDC matched 777777577 factored 999999599 factored 777777677 factored 333331333 factored 999992999 NF 86, P-1 NF; PRP C, CERT 111113111 factored 333334333 factored 777776777 NF 85, P-1 NF, PRP C, CERT 777767777 factored 111181111 factored 111191111 factored 333233333 factored 999299999 NF 86, P-1 NF, PRP C, CERT 999499999 factored 999599999 NF 86, P-1 NF, PRP C, CERT 333733333 factored 333833333 factored 115111111 factored 776777777 factored 337333333 NF 81; P-1 NF, PRP C, CERT 118111111 LL C; LLDC matched 778777777 factored 998999999 NF 86; P-1 NF, PRP C, CERT 101111111 factored 131111111 factored 373333333 factored 787777777 factored 577777777 factored 799999999 NF 85; P-1 NF; PRP C, CERT Counted 33 (end) 23 factored, 2 LL & 8 PRP composite primality test results, 0 remaining to be determined in the list immediately above. (Bold purple font above indicates apparently stalled assignments; blue italic runs in progress by Kriesel; underlined unassigned, oked by PM by Prime95.) There's also a very short very old forum thread about such numbers. Ten-digit near-repdigits Code: # nearrep3.pl # perl script to find 10-digit near repdigit prime exponents i..iji..i, j != i, i>0, base 10 # where leftmost digit may be j, some middle digit may be, or rightmost may be # this script spins through many i,j,l(position of j) for simplicity at cost of efficiency use ntheory; print "nearrep3.pl (C) 2022 Kriesel, generates list of 10-digit base 10 near-repdigit primes.\n"; print "1234567890\n----------\n";$count=0; for ( $i=0;$i<10; $i++ ) { # repfield is$i as digits x 9 places for ( $l=0;$l<10; $l++ ) { # power of ten at which digit differs for ($j=0; $j<10;$j++ ) { # differing digit at any position if ( $i !=$j ) { $k=$i*1111111111 +($j-$i) *10**$l; if ( Math::Prime::Util::GMP::is_prime($k) == 0 ) { # print "$k is composite\n"; } elsif ($k > 1e9 ) { # skip the leading-0 case which will give 9-digit rep digit print "$k\n";$count++; } # else { print "$k is too small\n"; } } # else { print "i=j=$j\n"; } } } } print "Counted \$count\n"; An annotated and hand-sorted output list follows, with link encoding: 1111111121 factored 1111111181 factored 1111111411 factored 1111115111 factored 1111211111 factored 1111411111 factored 1115111111 factored 1117111111 factored 1121111111 factored 1151111111 factored -----prime95 / AVX512 64M fft limit----- 1711111111 factored 1777777777 factored -----gpuowl v6.11-380/7.x / 120M fft limit----- 2777777777 TF 90 NF = TF goal 3233333333 factored -----gpuowl v6.5-84 / 192M fft limit------ 3333133333 factored 3333323333 factored 3333332333 TF 89 NF, goal 91 3333333323 factored 3333333833 factored 3334333333 factored -----mfaktx / unsigned 32-bit exponent limit----- 4444444447 factored 5555555557 TF 80 NF, goal 94 6666666661 factored 7727777777 TF 80 NF, goal 95 7777717777 TF 80 NF, goal 95 7777747777 TF 80 NF, goal 95 7777772777 TF 79 NF, goal 95 7777777577 factored 7778777777 TF 79 NF, goal 95 8777777777 factored -----mlucas V20.1.1 / 512M fft limit----- 9199999999 factored 9299999999 factored 9959999999 TF 79 NF, goal 96 9995999999 factored 9999499999 TF 79 NF, goal 96 9999929999 TF 79 NF, goal 96 (76 to 78 was 3.5 days on one core of i5-1035G1 in mfactor; other exponents will be longer) 9999959999 factored 9999999929 factored Counted 38 Of those, 11 have survived mostly incomplete TF so far, and TF continues in mfaktc for the unfactored exponent within range of mfaktx, and in Ernst's mfactor on a single core continues intermittently for the unfactored exponents too large for mfaktx. These surviving 10-digit near-repdigit exponent mersenne numbers are all beyond the reach of prime95 and of plausible primality test times. P-1 factoring them adequately would require considerable ram and long run times, and in a few cases extension of Mlucas to somewhat larger fft lengths. Not all TF bit levels completed are reported to mersenne.ca yet. Taking those beyond mfaktx capability from 74 to 76 bits was ~1 day each in Mfactor on a single core of an i5-1035G1 system using a single thread, and 79 to 80 ~12 days each. Mfactor can be run in highly parallel form, one core per class, so given a sufficient number of cores available, acceleration can be considerable by running many processes in parallel (as 2, 4, 8 or 16 threads, or any factor of 960). However, mfactor currently does not support save files, so recovery from interrupted runs is manual, which becomes burdensome at high thread counts. Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1 Last fiddled with by kriesel on 2023-05-30 at 10:50 Reason: status update
 2020-12-24, 18:10 #5 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 11110001100102 Posts Palindromic numbers as exponents Palindromic numbers are numbers that are reversible digit by digit without changing value. 110505011 is an example, which as a Mersenne exponent, leads to a P-1 factor. Palindromic numbers in base ten are quite common. The subset that are primes are also common. There are 5172 in 108
2020-12-24, 18:18   #6
kriesel

"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

773010 Posts
Prime approximations of round multiples of irrational or transcendental numbers

Consider [ceiling|floor] r = c x 10n x b, where b is an irrational or transcendental number such as pi, e, sqrt(2), sqrt(3), sqrt(5), sqrt(10), etc and where c, n are are small positive integers chosen such that 108< r < 109. And find the nearest prime exponent to r. Or [ceiling|floor] r = c x 10n x bm where b is a transcendental number and m is also a small positive integer.

Factoring or primality testing on such exponents may be useful by falling in regions of the number line that would not be selected otherwise in a simple QA selection method.

Some early examples of such runs revealed limitations on CUDAPm1 exponent that were later found to be partly GPU-model dependent.
130637753 (factored) < 108 Mills' constant ~ 130637788 < 130637791 (PRP C & cert)
141421333 (PRP C & cert) < 108 sqrt (2) ~ 141421356 < 141421387 (factored)
144818161 (PRP C & cert) < 108 sum of reciprocals of known Mersenne primes' exponents ~ 144818186 < 144818221 (PRP C & cert; this line added 2021-08-18)
147576139 (factored) < 108 2^(1/egamma) ~ 147576140 < 147576181 (factored)
157079621 (PRP C & cert) < 108 pi/2 ~ 157079632.7 < 157079633 (factored)
161803393 (factored) < 108 golden ratio ~ 161803399 < 161803403 (PRP C & cert)
173205079 (PRP C & cert) < 108 sqrt (3) ~ 173205081 < 173205089 (factored)
192878201 (factored) < 108 Wright's constant (= 1.928782187...) ~ 192878219 < 192878237 (factored)
223606793 (PRP C & cert) < 108 sqrt (5) ~ 223606798 < 223606807 (PRP C & Cert)
261497207 (factored) < 109 Mertens constant ~ 261497213 < 261497239 (PRP C & Cert)
271828171 (PRP C & cert) < 108 e ~ 271828183 < 271828199 (PRP C & cert)
292005097 (factored) < 108 "Buenos Aires constant" ~ 292005098 < 292005113 (PRP C & cert)
314159257 (PRP C & cert) < 108 pi ~ 314159265 < 314159311 (factored)
316227731 (factored) < 108 sqrt(10) ~ 316227766 < 316227767 (factored)
543656363 (factored) < 2 x 108 e ~ 543656366 < 543656371 (PRP C; Cert mismatch twice; needs PRPDC)
577215631 (TF & P-1 done, PRP C, needs PRP DC) < 109 Euler-Mascheroni constant ~ 577215665 < 577215677 (factored)

(PrimeNet server limit for automatic processing of proof files is up to ~596M)

624329977 (factored) < 109 Golomb-Dickman constant ~ 624329989 < 624330017 (factored)
628318517 (TF & P-1 done, PRP needed) < 2 x 108 pi ~ 628318531 < 628318583 (TF & P-1 done, PRP needed)
785398129 (factored) < 109 pi/4 ~ 785398163 < 785398169 (factored)
853973387 (factored) < 108 e pi ~ 853973422 < 853973437 (TF & P-1 done, PRP needed)
942477787 (factored) < 3 x 108 pi ~ 942477796 < 942477799 (TF done, P-1 done, PRP needed)
986960431 (factored) < 108 pi2 ~ 986960440 < 986960461 (TF & P-1 done, PRP needed)
Suggestions for additions to the above list are welcome by PM.
(purple font identifies active assignments; green font identifies assignments available)

Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1
Attached Files
 log math constants and benfords law.pdf (23.1 KB, 117 views)

Last fiddled with by kriesel on 2023-03-15 at 14:04 Reason: status update

 2020-12-24, 21:23 #7 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 2×5×773 Posts Personally significant dates encoded into exponents The idea of encoding personally significant dates into exponents came up in https://www.mersenneforum.org/showpo...04&postcount=8 and has no more mathematical merit than trying to pick a winning lottery ticket by the same method, which is no merit at all. But if it's fun, and you're not concerned about the leakage of personally identifying information, try it. Birthdates, anniversaries, etc. Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1 Last fiddled with by kriesel on 2021-01-03 at 14:42
 2022-04-21, 01:38 #9 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 2·5·773 Posts Dual digit value palindromic exponents Consider some palindromic prime exponents with cyclic digit patterns possible with exactly two distinct digit values, a and b, both >0, for 9-digit base 10 exponents, with digit cycle periods, and excluding near-repdigits such as aaaabaaaa: (1 N/A; repdigit, all of which are composite) 2 ababababa 3 abaabaaba (oops, that is a repdigit too, in base 1000, so composite) 4 abbbabbba 5 abbaaabba (6 aaabbbaaa is always divisible by 3; 111=3 x 37) 7 aabbbbbaa 8 abbbbbbba For the multidigit exponent to be prime in base 10, the rightmost digit a must be 1, 3, 7 or 9. (And for a palindrome, so must the leftmost digit.) Dr. Sardonicus helpfully contributed the following 6 PARI/GP code and output sections by a PM (while this post was in early draft form; edited slightly here, and posted with his permission): 2 ababababa Code: forstep(a=1,9,[2,4,2],for(b=0,9,n=a*(10^8+10^6+10^4+10^2+1)+b*(10^7+10^5+10^3+10);if(gcd(n,3*7*11*13*17)==1&&ispseudoprime(n),print(n)))) 323232323 383838383 727272727 919191919 929292929 979797979 989898989 3 abaabaaba Code: forstep(a=1,9,[2,4,2],for(b=0,9,n=a*(10^8+10^6+10^5+10^3+10^2+1)+b*(10^7+10^4+10);if(gcd(n,3*7*11*13*17)==1&&ispseudoprime(n),print(n)))) ? /*Note: aba*(10^6+10^3+1)*/ 4 abbbabbba Code: forstep(a=1,9,[2,4,2],for(b=0,9,n=a*(10^8+10^4+1)+b*(10^7+10^6+10^5+10^3+10^2+10);if(gcd(n,3*7*11*13*17)==1&&ispseudoprime(n),print(n)))) ? /*Note: always divisible by 3*/ 5 abbaaabba Code: forstep(a=1,9,[2,4,2],for(b=0,9,n=a*(10^8+10^5+10^4+10^3+1)+b*(10^7+10^6+10^2+10);if(gcd(n,3*7*11*13*17)==1&&ispseudoprime(n),print(n)))) 133111331 377333773 766777667 944999449 977999779 988999889 7 aabbbbbaa Code: forstep(a=1,9,[2,4,2],for(b=0,9,n=a*(10^8+10^7+10+1)+b*(10^6+10^5+10^4+10^3+10^2);if(gcd(n,3*7*11*13*17)==1&&ispseudoprime(n),print(n)))) 331111133 772222277 779999977 992222299 995555599 8 abbbbbbba Code: forstep(a=1,9,[2,4,2],for(b=0,9,n=a*(10^8+1)+b*(10^7+10^6+10^5+10^4+10^3+10^2+10);if(gcd(n,3*7*11*13*17)==1&&ispseudoprime(n),print(n)))) 188888881 199999991 322222223 355555553 722222227 Checking status of each of the resulting 23 exponents, yields 20 shown composite, 0 needing Cert, 3 still pending determination, as follows: 323232323 TF NF; P-1 NF; LL C; PRP C & Cert 383838383 TF NF; P-1 NF; LL C; LLDC C match 727272727 TF NF; P-1 NF; no primality test 919191919 3 known factors 929292929 known factor 979797979 known factor 989898989 TF NF; P-1 NF; no primality test 133111331 2 known factors 377333773 2 known factors 766777667 2 known factors 944999449 2 known factors 977999779 known factor 988999889 known factor 331111133 known factor 772222277 2 known factors 779999977 known factor 992222299 TF NF; P-1 NF; no primality test 995555599 known factor 188888881 TF NF; P-1 NF; LL C; LLDC C match 199999991 2 known factors 322222223 TF NF; P-1 NF; LL C; PRP/proof as DC & CERT 355555553 known factor 722222227 3 known factors Here black normal font means completed; black bold means current assigned to someone else; green bold indicates available; blue italic indicates being run by kriesel; underlined indicates without a reservation (probably one could not be obtained); bold purple indicates stalled old assignments Are there more? yes; 4 aabaaabaa 5 abbababba 5 aaaabaaaa 6 aaababaaa 7 aabbabbaa 8 abaaaaaba 8 ababbbaba yielding respectively, 110111011 known factor 112111211 TF NF; P-1 NF; LL C & LLDC C match 113111311 TF NF; P-1 NF; LL C, PRP C & CERT 115111511 2 known factors 331333133 known factor 335333533 known factor 338333833 TF NF; P-1 NF; PRP C & CERT 991999199 known factor 322323223 4 known factors 355353553 TF NF P-1 NF; PRP C & CERT 722727227 TF NF; P-1 NF; no primality test 911919119 3 known factors 111181111 known factor 111191111 3 known factors 777767777 2 known factors 111010111 known factor 111515111 2 known factors 111616111 2 known factors 333434333 TF NF; P-1 NF; PRP C & CERT 333535333 known factor 112212211 known factor 118818811 2 known factors 338838833 known factor 994494499 known factor 998898899 TF NF; P-1 NF; no primality test 121111121 2 known factors 131111131 TF NF; P-1 NF; PRP C & CERT 181111181 TF NF; P-1 NF; PRP C & CERT 323333323 TF NF; P-1 NF; PRP C & CERT 131333131 TF NF; P-1 NF; PRP C & CERT 181888181 known factor 313111313 known factor 323222323 known factor 383888383 TF NF; P-1 NF; PRP C & CERT 959555959 known factor of the 35, 33 are shown composite, 2 are yet to be determined, & 0 need DC or CERT. So overall above, 58, of which 53 are shown composite, 5 are yet to be determined, and 0 need DC or CERT. Are there more? Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1 Last fiddled with by kriesel on 2023-04-11 at 17:54 Reason: style and formatting cleanup
 2022-08-07, 17:48 #10 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 2×5×773 Posts Near-palindromic exponents Near palindromes differ by one character from a palindrome. These will be more numerous than palindrome exponents. All the following concerns base-ten exponent representations. Consider an arbitrary palindromic exponent abcdeDCBA. For a palindrome a=A, b=B, c=C, d=D. Any of bcde might have value any digit 0-9. A will be only 1 3 7 or 9 or the exponent would be composite. For a near palindrome exponent, any one of bcdDCB may have any of the other 9 values than for the palindrome, while a may have any of the other 8 excluding 0, and A may have one of three other values and still possibly result in a prime exponent. So it appears 9-digit near palindrome exponents would be about 8+6*9+3 = ~65 times more numerous than palindromes. There are ~5172 nine digit prime palindrome exponents. Nine-digit prime near palindromes may number around 336,180. The above simply derived estimate produces an overcount, an upper bound. Consider these prime palindromes: 100060001 101060101 Either might produce the same near palindromes by changing a single digit in the c or C position, 100060101 or 101060001, and these each get counted twice. Also for 103060301, 106060601, and no doubt many other cases. This applies equally to the b / B or d / D positions. The a / A positions are not as symmetric so not a simple factor of two there. For palindromes: eDCBA with A constrained to 4 values (1 3 7 9) gives 4E4 possible numbers, (abcd matching ABCD respectively), of which 5172 are prime (~12.93%) For abcdeDCBA near palindromes: Consider two cases; interior single symmetric digit mismatch, and exterior (leading/trailing digit) mismatch. Interior: with A constrained to 4 values (1 3 7 9) and a=A gives 1E9 x 0.4 * 0.1 = 4E7 possible numbers. Constraining any 2 of bcd to respectively match BCD reduces the possible numbers by a factor of 3/100 to 1.2E6. And constraining the remaining one of bcd to not match its counterpart provides a further factor of 0.9, to 1.08E6. 4 values a,A x 10 values e x 1000 values bcd x 3*9 BCD single digit mismatches to bcd = 1,080,000. Check. Exterior: For a!=A, for each allowable A to possibly produce a prime there are 8 possible a values >0 & != A; and all of bcd must match their corresponding digits DCB; 1E9 * 0.8 * 0.4 * 0.001 = 320,000. 8 possible values a >0 !=A, x 4 A, x 10,000 bcde = 320,000. Check. The two cases a=A and a!=A are mutually exclusive, so the numbers can be summed. Combining cases, 1,400,000, of which likely ~0.1293 are prime; ~181020. This is ~1.0357 times higher (+6245) than 174775 that axn obtained with a Pari script counting 9-digit near palindrome primes. Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1 Last fiddled with by kriesel on 2022-08-08 at 18:55
 2023-01-13, 23:14 #11 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 11110001100102 Posts Double Mersennes and higher A Mersenne number with a mersenne prime as its exponent is called a double Mersenne. Triple and higher are also possible. 1: M2 =3; M3= 7; M5=31; M7=127; 2: MM2 = 2^(2^2 -1) - 1 = M3 = 7; MM3 = M7 = 127; MM5 = M31 = 2,147,483,647; MM7 = M127 = 170141183460469231731687303715884105727 3: MMM2 = MM3 = M7; MMM3 = MM7 = M127; MMM5 = MM31 = M2147483647 4: MMMM2 = MMM3 = MM7 = M127 = 170141183460469231731687303715884105727; MMMM3 = MMM7 = MM127 = M170141183460469231731687303715884105727 5: MMMMM2 = MMMM3 = MMM7 = MM127 = M170141183460469231731687303715884105727 6 or higher: none known MM31 = M2147483647 has known factors. (Therefore MMM31 is not a triple mersenne.) It's unknown whether MM61, MM89, MM107, or MM127 are prime. There's a web site featuring these at http://www.doublemersennes.org/ Except for MM31, they are beyond reach of PRP or LL based primality testing, or P-1 or ECM factoring. Trial factoring is feasible via modpow, as described in the TF section of https://www.mersenne.org/various/math.php. MM31 and larger are too large for prime95's P+1 factoring capability, which is limited to 1169M exponent on AVX512 hardware, & lower on other hardware. MM31 could be P-1 factored in some versions of gpuowl on rare GPUs with ~40GB of ram or more for stage 2, or in Mlucas v20.x on systems with sufficient ram (~48GiB or more, more is better for stage 2). Double mersennes up to MM31 can be trial factored in mfaktc to 95 bits, or 92 bits in mfakto. The mmff application was created specifically to trial factor MM31, MM61, MM89, MM107, or MM127 on NIVIDIA GPUs. Ernst's mfactor program and Luigi's Factor5 can also be used, at higher bit levels, but should not be used in situations where GPU applications can be used. From limited testing, it appears mfaktc is faster than mmff (~2:1) on MM31 on the same hardware. Mmff supports dividing up bit levels among multiple runs/GPUS/users by specifying start and end k values in the worktodo entry. Mfaktc does not support that syntax, supporting as input only integer bit levels. There's a collaborative trial factoring project for MM31, MM61, MM89, MM107, and MM127. Reservations are made in a thread for them, https://mersenneforum.org/showthread...=17186&page=33 and results are reported similarly at https://mersenneforum.org/showthread...=17187&page=40 https://mersenneforum.org/showthread.php?t=17162 is the thread for development and support of mmff. If I read the v0.28 mmff source correctly, TF on MM127 is supported up to factors 2188. Currently, 2023-06-07, gapless TF has reached the following levels: Code: MMp URL k max done ~bits kernels’ max bits bits left MM31 http://www.doublemersennes.org/mm31.php 1153.E15 92.00 89, 96 4.00 MM61 http://www.doublemersennes.org/mm61.php 270.E15 119.91 108, 120, 128 8.09 MM89 http://www.doublemersennes.org/mm89.php 58.E15 145.69 128, 140, 152, 160 14.31 MM107 http://www.doublemersennes.org/mm107.php 10.E15 161.15 152, 160, 172 10.85 * MM127 http://www.doublemersennes.org/mm127.php 190.E15 185.40 183, 185, 188 2.60 * * some higher k blocks have also been completed. If the currently completed and reported TF had been performed without gaps, they would reach to approximately: Code: p equivalent k depth ~bits equiv bits left M107 11E15 161.29 10.71 M127 395E15 186.45 1.55 The bits values above are calculated from effective largest factor tried as follows. for Mp, f=2kp+1; for large f ~2kp, so for MMp, f~2 Mp k, log2 (f) ~ log2(2) + log2(Mp) + log2(k); for MM127: f~2 M127 k; log2(f) ~1 + 127 + log2(k) = 128 + log(380e15)/log(2) = 186.40 For MM107: log2(f) ~ 1 + 107 + log2(11E15) ~ 161.29 Only four double mersennes are known to be prime; MM2 = 7, MM3 = 127; MM5 = 2147483647; MM7 = 170141183460469231731687303715884105727. Only four are known to be composite; MM13 = M8191, MM17 = M131071, MM19 = M524287, MM31 = M2147483647. The rest are undetermined. It's thought none past MM7 are prime. There has also been a limited amount of trial factoring applied to the higher exponent double Mersennes derived from all the currently known Mersenne primes. See http://www.doublemersennes.org/history.php These are difficult to factor, with a trial factoring with a single well chosen candidate factor constituting about as much work as a primality test of the exponent; that is, for MM82589933, one TF trial is about as much computational work as a PRP or LL test on M82589933. Top of reference tree: https://www.mersenneforum.org/showpo...22&postcount=1 Last fiddled with by kriesel on 2023-06-07 at 14:41 Reason: updated for MM127 progress

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