20090711, 19:26  #23 
May 2009
Loughborough, UK
2^{2}×11 Posts 
axn is of course right.
Also 2*43112609 does divide 2*(2^{43112608}1) 43112609 first divides 2^{10778152}1 but I doubt it has the smallest k for that exponent. Last fiddled with by plandon on 20090711 at 19:30 Reason: brackets added 
20090711, 20:14  #24 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{2}·7·389 Posts 

20090711, 20:45  #25 
Einyen
Dec 2003
Denmark
2^{2}×859 Posts 
Sorry, instead of thinking logically I tried to test it numerically and misscalculated somehow.

20090712, 01:01  #26  
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
11×389 Posts 
Quote:
Code:
(20:00) gp > Mod(2^431126092,2*43112609) %2 = Mod(0, 86225218) (20:00) gp > Mod(2^426438012,2*42643801) %3 = Mod(0, 85287602) Last fiddled with by TimSorbet on 20090712 at 01:07 

20090712, 01:18  #27 
Jun 2003
153C_{16} Posts 
Ok. What _is_ the smallest factor of a Mersenne prime?
There is nothing "seems to" about it. Hint: Fermat's Little Theorem 
20090712, 01:29  #28  
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
10267_{8} Posts 
I meant this following point, not this thread's focus:
Quote:
Last fiddled with by TimSorbet on 20090712 at 01:33 

20090712, 01:37  #29 
Jun 2003
2^{2}×3^{2}×151 Posts 
So did I (as explained in Post #22 second point). Which is why asked: what is the smallest factor of a mersenne prime? Do you consider it as the Mersenne prime itself? Or do you consider it as having no factor (for the purpose of this thread)?
Last fiddled with by axn on 20090712 at 01:38 
20090712, 02:13  #30 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
11·389 Posts 
Oh, I misunderstood and thought you were talking about Mersenne numbers and not Mersenne primes. I just misread.
I'd say no...I don't have any particularly good reason for this, just that then this thread would have a trivial answer: (Mp1)/2p where p is the exponent of the current largest known Mersenne prime (currently 43112609). 
20090712, 02:24  #31  
Jun 2003
2^{2}·3^{2}·151 Posts 
Quote:
EDIT: btw, I agree with your sentiments on post #8 Last fiddled with by axn on 20090712 at 02:34 

20090712, 14:45  #32 
Einyen
Dec 2003
Denmark
2^{2}·859 Posts 
I got curious about kvalues so I made a program find factors 2kp+1 with fixed k for p up to 1 billion (like primenet v5). There is 50,847,534 primes up to 1 billion minus the 47 known mersenne primes, so 50,847,487 primes left.
I discovered k=+/2 (mod 8) gives no factors. This comes from the fact that 2kp+1 has to be +/ 1 (mod 8): http://primes.utm.edu/notes/proofs/MerDiv.html 2kp + 1 = +1/1 (mod 8) => 2kp = 0/2 (mod 8) (this is same as 6/8) => kp = 0/3/4/7 (mod 8). Primes p is always 1,3,5 or 7 mod 8, so: p=1 (mod 8): kp = 0/3/4/7 => k=0/3/4/7 (mod 8) p=3 (mod 8): kp = 0/3/4/7 => k=0/1/4/5 (mod 8) p=5 (mod 8): kp = 0/3/4/7 => k=0/3/4/7 (mod 8) p=7 (mod 8): kp = 0/3/4/7 => k=0/1/4/5 (mod 8) So k=2/6 (mod 8) gives no factors. Column A is the number of mersennenumbers 2^{p}1 with p<=10^{9} where 2kp+1 is the smallest factor. Column B is the number of mersennenumbers 2^{p}1 with p<=10^{9} where 2kp+1 is a factor, not necessarily the smallest. I left out B when it was same as A. Code:
A B k=1 1655600 k=2 0 k=3 773434 k=4 409773 442260 k=5 191162 k=6 0 k=7 83083 88872 k=8 109830 114789 k=9 81930 90882 k=10 0 k=11 32549 34274 k=12 94461 104326 k=13 20504 24545 k=14 0 k=15 40699 44859 k=16 24778 29716 k=17 12909 13999 k=18 0 k=19 9752 11171 k=20 23567 25772 k=21 17568 21032 k=22 0 k=23 6754 7636 k=24 23341 27013 k=25 6584 8409 k=26 0 k=27 9483 10696 k=28 10005 12160 k=29 4370 4839 k=30 0 k=31 3714 4323 k=32 6909 7716 k=33 6793 8161 k=34 0 k=35 4617 5208 k=36 10360 12435 k=37 2340 3022 k=38 0 k=39 4925 5708 k=40 5310 6643 k=41 2174 2420 k=42 0 k=43 1947 2311 k=44 4037 4573 k=45 4316 5315 k=46 0 k=47 1633 1850 k=48 5954 7180 k=49 1472 1990 k=50 0 k=51 2860 3337 k=52 2640 3326 k=53 1319 1493 k=54 0 k=55 1627 1956 k=56 2724 3187 k=57 2148 2681 k=58 0 k=59 1089 1275 k=60 5043 6193 k=61 774 1133 k=62 0 k=63 2030 2444 k=64 1537 2009 k=65 1210 1427 k=66 0 k=67 762 937 k=68 1566 1845 k=69 1393 1805 k=70 0 k=71 770 894 k=72 2601 3214 k=73 576 800 k=74 0 k=75 1668 2014 k=76 1164 1523 k=77 770 899 k=78 0 k=79 544 696 k=80 1505 1794 k=81 994 1300 k=82 0 k=83 510 608 k=84 2317 2870 k=85 571 829 k=86 0 k=87 990 1226 k=88 959 1241 k=89 464 549 k=90 0 k=91 499 622 k=92 884 1076 k=93 807 1020 k=94 0 k=95 562 688 k=96 1468 1852 k=97 325 440 k=98 0 k=99 812 989 k=100 874 1160 k=101 339 410 k=102 0 k=103 329 415 k=104 730 858 k=105 940 1218 k=106 0 k=107 276 324 k=108 1116 1482 k=109 252 369 k=110 0 k=111 623 755 k=112 614 811 k=113 307 381 k=114 0 k=115 352 451 k=116 507 615 k=117 537 683 k=118 0 k=119 293 365 k=120 1170 1523 k=121 228 329 k=122 0 k=123 474 587 k=124 409 577 k=125 321 381 k=126 0 k=127 212 272 k=128 428 529 k=129 417 521 k=130 0 k=131 243 292 k=132 838 1088 k=133 220 324 k=134 0 k=135 527 662 k=136 378 509 k=137 217 247 k=138 0 k=139 171 214 k=140 603 734 k=141 320 456 k=142 0 k=143 172 230 k=144 617 785 k=145 197 299 k=146 0 k=147 356 470 k=148 276 389 k=149 138 178 k=150 0 k=151 155 203 k=152 337 413 k=153 324 429 k=154 0 k=155 192 237 k=156 594 761 k=157 126 176 k=158 0 k=159 272 346 k=160 341 460 k=161 169 212 k=162 0 k=163 138 175 k=164 258 328 k=165 373 512 k=166 0 k=167 117 147 k=168 585 743 k=169 124 175 k=170 0 k=171 221 284 k=172 202 289 k=173 131 155 k=174 0 k=175 184 228 k=176 252 315 k=177 205 290 k=178 0 k=179 117 140 k=180 511 680 k=181 81 122 k=182 0 k=183 204 270 k=184 193 266 k=185 131 163 k=186 0 k=187 107 151 k=188 205 251 k=189 219 294 k=190 0 k=191 99 123 k=192 386 507 k=193 73 117 k=194 0 k=195 278 344 k=196 187 275 k=197 91 117 k=198 0 k=199 94 119 k=200 223 284 k=201 171 241 k=202 0 k=203 100 135 k=204 354 454 k=205 113 162 k=206 0 k=207 179 229 k=208 175 235 k=209 122 144 k=210 0 k=211 75 95 k=212 164 202 k=213 143 192 k=214 0 k=215 105 137 k=216 288 384 k=217 73 101 k=218 0 k=219 170 218 k=220 204 291 k=221 81 99 k=222 0 k=223 68 84 k=224 154 201 k=225 173 231 k=226 0 k=227 61 85 k=228 269 352 k=229 66 93 k=230 0 k=231 167 208 k=232 148 202 k=233 76 92 k=234 0 k=235 66 89 k=236 147 190 k=237 108 155 k=238 0 k=239 75 88 k=240 304 407 k=241 64 93 k=242 0 k=243 104 140 k=244 119 168 k=245 100 118 k=246 0 k=247 71 90 k=248 117 148 k=249 102 151 k=250 0 k=251 46 61 k=252 261 352 k=253 48 69 k=254 0 k=255 157 200 Total 3790227 Last fiddled with by ATH on 20090712 at 14:47 
20090712, 15:34  #33 
Nov 2008
2×3^{3}×43 Posts 
1 is the smallest factor of any whole number.

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