Are Mersenne numbers really square?

[Dr Sardonicus]

Originally Posted by GP2

It's not the exponent that would be the next Wieferich prime, but the non-square-free factor itself.

Oops!

Right. So, q² divides 2^q-1 - 1, and the multiplicative order of 2 (mod q²) is a prime divisor p of q - 1. This is not true of the two known Wieferich primes to the base 2:

The multiplicative order of 2 (mod 1093²) is 1092/3 = 346, and the multiplicative order of 2 (mod 3511²) is 3510/2 = 1755. Neither 346 nor 1755 is prime.

However, to the base 3, we have the small example

3⁵ - 1 = 242 = 2 * 11², so that

3⁵ == 1 (mod 11²).

[JM Montolio A]

I tell us a secret. Dont waste time looking for. They are all square free.
The Mersenne numbers, the Wagstaff numbers, and the Fermat numbers.
That follows of the properties of the M() function.

[science_man_88]

We get for free that if they aren't then the divisor that opposes the square are 7 mod 8. Doesn't help much though.edit: okay it limits the number of arrangements to n for 2n+1 7 mod 8 divisors.

[Dr Sardonicus]

Originally Posted by JM Montolio A

I tell us a secret. Dont waste time looking for. They are all square free.
The Mersenne numbers, the Wagstaff numbers, and the Fermat numbers.
That follows of the properties of the M() function.

Unfortunately, you have yet to completely define M(p) for p prime.

In the Initial post to the thread A useful function. you wrote:

Define M(n) as:


for (p^e), M( p^e ) = M(p)*(p ^ (e-1) )


for (m,n ) coprimes, M(n*m)= (M(n)*M(m))/(mcd(M(n),M(m))


for p prime, p | (2^M(p)-1)

In Post #8 you added

for p prime, M(p)|(p-1)

and in Post #11 you added

other property, M( 2^e - 1 ) = e.

As far as I can tell, for p prime, M(p) is only well defined generally when either (a) p is a Mersenne prime, or (b) 2 is a primitive root (mod p). In Post #15 you also individually defined the specific values

M( 1093 ) = 364

M( 3511 ) = 1755

I also note that, due to your condition

for (p^e), M( p^e ) = M(p)*(p ^ (e-1) )

quoted above, if p is a Wieferich prime, your M(p^e) for e > 1 is no longer the multiplicative order of 2 (mod p^e). For example, 346 is the multiplicative order of 2 (mod 1093) and also (mod 193^2).

So your M(1093^2) = 1093*346 is the multiplicative order of 2 (mod 1093^3).

[JM Montolio A]

I think axiomatic definition for M() is enough, but talking on code approach, I post the pseudocode on:

http://www.mersenneforum.org/showpos...7&postcount=19

I try to post also the .exe, but i find a security rule.

Can be "The Staff" can leave me to post here the full C-code of M(). A few lines of C for 64 bits n.

Thanks to all for your interest on my M() function.
Really I think is something like a good step.
And happy to share it here and now.

JM M

[JM Montolio A]

On the zip the C code.

gcc -o ..

[Dr Sardonicus]

Originally Posted by JM Montolio A

I think axiomatic definition for M() is enough

Pfui. In this post, I listed 20 primes p less than 200 for which your "definition" of M(p) was ambiguous. They could be any value divisible by the multiplicative order of 2 (mod p), and dividing p-1.

I repost the values here for ease of reference:

For p = 17, the possible values of M(p) are 8 times k for k in [1, 2].
For p = 23, the possible values of M(p) are 11 times k for k in [1, 2].
For p = 41, the possible values of M(p) are 20 times k for k in [1, 2].
For p = 43, the possible values of M(p) are 14 times k for k in [1, 3].
For p = 47, the possible values of M(p) are 23 times k for k in [1, 2].
For p = 71, the possible values of M(p) are 35 times k for k in [1, 2].
For p = 73, the possible values of M(p) are 9 times k for k in [1, 2, 4, 8].
For p = 79, the possible values of M(p) are 39 times k for k in [1, 2].
For p = 89, the possible values of M(p) are 11 times k for k in [1, 2, 4, 8].
For p = 97, the possible values of M(p) are 48 times k for k in [1, 2].
For p = 103, the possible values of M(p) are 51 times k for k in [1, 2].
For p = 109, the possible values of M(p) are 36 times k for k in [1, 3].
For p = 113, the possible values of M(p) are 28 times k for k in [1, 2, 4].
For p = 137, the possible values of M(p) are 68 times k for k in [1, 2].
For p = 151, the possible values of M(p) are 15 times k for k in [1, 2, 5, 10].
For p = 157, the possible values of M(p) are 52 times k for k in [1, 3].
For p = 167, the possible values of M(p) are 83 times k for k in [1, 2].
For p = 191, the possible values of M(p) are 95 times k for k in [1, 2].
For p = 193, the possible values of M(p) are 96 times k for k in [1, 2].
For p = 199, the possible values of M(p) are 99 times k for k in [1, 2].

So far, you have failed to resolve the ambiguity in any of them, let alone explain how this may be done.

I'm not a programmer. But from my limited experience with computer programs, being able to explain the algorithm is easier than -- indeed, a prerequisite for -- writing the program.

[JM Montolio A]

Well. Step to step, Doc.
I think M() can have one full correct axiomatic definition. BUT ALSO I post the pseudocode, at other forum. And the C code here. The theory of the M function start solving the question:¿ whats the distance from a prime p to any f(i) series?. Answer: f(d) mod p. After i find what i named "t-serie". Here is "n+e=(2^g)e'". Any t-serie gives a integral equation. Here is: (eEnd)(2^M)-(eStart) = n*D.

The C source is only the resolution of the tserie.
This is, start n+1=(2^g)e',n+e'=(2^g)e'',until you get e=1.

¿ always you back to one ? Yes.
¿ is something like Collatz ending always on 1 ? Yes, is the same thing. I posted also about it.

If you compile, links, and execute the C code, you get the right results.

For a practical user, what makes the M() is put any prime on their place.
I think cpu time is best used computing M, that making trials on dividers.
The tserie is like the Touring Machine. Slow, but with a strong theory. Can be someone can make the step from my theorical tserie to a faster language.

JM M

[JM Montolio A]

I put some trace on the code.

n 3
n 3 + e 1 = (2^g 2)(e' 1)
n 3 M 2 Dbit 1 D 1

n 5
n 5 + e 1 = (2^g 1)(e' 3)
n 5 + e 3 = (2^g 3)(e' 1)
n 5 M 4 Dbit 2 D 3

n 7
n 7 + e 1 = (2^g 3)(e' 1)
n 7 M 3 Dbit 1 D 1

n 9
n 9 + e 1 = (2^g 1)(e' 5)
n 9 + e 5 = (2^g 1)(e' 7)
n 9 + e 7 = (2^g 4)(e' 1)
n 9 M 6 Dbit 3 D 7

n 11
n 11 + e 1 = (2^g 2)(e' 3)
n 11 + e 3 = (2^g 1)(e' 7)
n 11 + e 7 = (2^g 1)(e' 9)
n 11 + e 9 = (2^g 2)(e' 5)
n 11 + e 5 = (2^g 4)(e' 1)
n 11 M 10 Dbit 5 D 93

n 13
n 13 + e 1 = (2^g 1)(e' 7)
n 13 + e 7 = (2^g 2)(e' 5)
n 13 + e 5 = (2^g 1)(e' 9)
n 13 + e 9 = (2^g 1)(e' 11)
n 13 + e 11 = (2^g 3)(e' 3)
n 13 + e 3 = (2^g 4)(e' 1)
n 13 M 12 Dbit 6 D 315

n 15
n 15 + e 1 = (2^g 4)(e' 1)
n 15 M 4 Dbit 1 D 1

n 17
n 17 + e 1 = (2^g 1)(e' 9)
n 17 + e 9 = (2^g 1)(e' 13)
n 17 + e 13 = (2^g 1)(e' 15)
n 17 + e 15 = (2^g 5)(e' 1)
n 17 M 8 Dbit 4 D 15

n 19
n 19 + e 1 = (2^g 2)(e' 5)
n 19 + e 5 = (2^g 3)(e' 3)
n 19 + e 3 = (2^g 1)(e' 11)
n 19 + e 11 = (2^g 1)(e' 15)
n 19 + e 15 = (2^g 1)(e' 17)
n 19 + e 17 = (2^g 2)(e' 9)
n 19 + e 9 = (2^g 2)(e' 7)
n 19 + e 7 = (2^g 1)(e' 13)
n 19 + e 13 = (2^g 5)(e' 1)
n 19 M 18 Dbit 9 D 13797

n 21
n 21 + e 1 = (2^g 1)(e' 11)
n 21 + e 11 = (2^g 5)(e' 1)
n 21 M 6 Dbit 2 D 3

n 23
n 23 + e 1 = (2^g 3)(e' 3)
n 23 + e 3 = (2^g 1)(e' 13)
n 23 + e 13 = (2^g 2)(e' 9)
n 23 + e 9 = (2^g 5)(e' 1)
n 23 M 11 Dbit 4 D 89

n 25
n 25 + e 1 = (2^g 1)(e' 13)
n 25 + e 13 = (2^g 1)(e' 19)
n 25 + e 19 = (2^g 2)(e' 11)
n 25 + e 11 = (2^g 2)(e' 9)
n 25 + e 9 = (2^g 1)(e' 17)
n 25 + e 17 = (2^g 1)(e' 21)
n 25 + e 21 = (2^g 1)(e' 23)
n 25 + e 23 = (2^g 4)(e' 3)
n 25 + e 3 = (2^g 2)(e' 7)
n 25 + e 7 = (2^g 5)(e' 1)
n 25 M 20 Dbit 10 D 41943

n 27
n 27 + e 1 = (2^g 2)(e' 7)
n 27 + e 7 = (2^g 1)(e' 17)
n 27 + e 17 = (2^g 2)(e' 11)
n 27 + e 11 = (2^g 1)(e' 19)
n 27 + e 19 = (2^g 1)(e' 23)
n 27 + e 23 = (2^g 1)(e' 25)
n 27 + e 25 = (2^g 2)(e' 13)
n 27 + e 13 = (2^g 3)(e' 5)
n 27 + e 5 = (2^g 5)(e' 1)
n 27 M 18 Dbit 9 D 9709

n 29
n 29 + e 1 = (2^g 1)(e' 15)
n 29 + e 15 = (2^g 2)(e' 11)
n 29 + e 11 = (2^g 3)(e' 5)
n 29 + e 5 = (2^g 1)(e' 17)
n 29 + e 17 = (2^g 1)(e' 23)
n 29 + e 23 = (2^g 2)(e' 13)
n 29 + e 13 = (2^g 1)(e' 21)
n 29 + e 21 = (2^g 1)(e' 25)
n 29 + e 25 = (2^g 1)(e' 27)
n 29 + e 27 = (2^g 3)(e' 7)
n 29 + e 7 = (2^g 2)(e' 9)
n 29 + e 9 = (2^g 1)(e' 19)
n 29 + e 19 = (2^g 4)(e' 3)
n 29 + e 3 = (2^g 5)(e' 1)
n 29 M 28 Dbit 14 D 9256395

n 31
n 31 + e 1 = (2^g 5)(e' 1)
n 31 M 5 Dbit 1 D 1

[JM Montolio A]

"so far, failed,.... Dr. Sarcasticus."


n 3 M 2 Dbit 1 D 1
n 5 M 4 Dbit 2 D 3
n 7 M 3 Dbit 1 D 1
n 9 M 6 Dbit 3 D 7
n 11 M 10 Dbit 5 D 93
n 13 M 12 Dbit 6 D 315
n 15 M 4 Dbit 1 D 1
n 17 M 8 Dbit 4 D 15
n 19 M 18 Dbit 9 D 13797
n 21 M 6 Dbit 2 D 3
n 23 M 11 Dbit 4 D 89
n 25 M 20 Dbit 10 D 41943
n 27 M 18 Dbit 9 D 9709
n 29 M 28 Dbit 14 D 9256395
n 31 M 5 Dbit 1 D 1
n 33 M 10 Dbit 5 D 31
n 35 M 12 Dbit 5 D 117
n 37 M 36 Dbit 18 D 1857283155
n 39 M 12 Dbit 4 D 105
n 41 M 20 Dbit 10 D 25575
n 43 M 14 Dbit 7 D 381
n 45 M 12 Dbit 5 D 91
n 47 M 23 Dbit 9 D 178481
n 49 M 21 Dbit 10 D 42799
n 51 M 8 Dbit 2 D 5
n 53 M 52 Dbit 26 D 84973577874915
n 55 M 20 Dbit 8 D 19065
n 57 M 18 Dbit 9 D 4599
n 59 M 58 Dbit 29 D 4885260612740877
n 61 M 60 Dbit 30 D 18900352534538475
n 63 M 6 Dbit 1 D 1
n 65 M 12 Dbit 6 D 63
n 67 M 66 Dbit 33 D 1101298153654301589
n 69 M 22 Dbit 11 D 60787
n 71 M 35 Dbit 14 D 483939977
n 73 M 9 Dbit 3 D 7
n 75 M 20 Dbit 9 D 13981
n 77 M 30 Dbit 15 D 13944699
n 79 M 39 Dbit 17 D 6958934353
n 81 M 54 Dbit 27 D 222399981598543
n 83 M 82 Dbit 41 D 0
n 85 M 8 Dbit 2 D 3
n 87 M 28 Dbit 11 D 3085465
n 89 M 11 Dbit 4 D 23
n 91 M 12 Dbit 4 D 45
n 93 M 10 Dbit 3 D 11
n 95 M 36 Dbit 14 D 723362913
n 97 M 48 Dbit 24 D 2901803883615
n 99 M 30 Dbit 15 D 10845877
n 101 M 100 Dbit 50 D 0
n 103 M 51 Dbit 23 D 21862134113449
n 105 M 12 Dbit 4 D 39
n 107 M 106 Dbit 53 D 0
n 109 M 36 Dbit 18 D 630453915
n 111 M 36 Dbit 14 D 619094385
n 113 M 28 Dbit 14 D 2375535
n 115 M 44 Dbit 19 D 152975530821
n 117 M 12 Dbit 3 D 35
n 119 M 24 Dbit 9 D 140985
n 121 M 110 Dbit 55 D 0
n 123 M 20 Dbit 6 D 8525
n 125 M 100 Dbit 50 D 0
n 127 M 7 Dbit 1 D 1
n 129 M 14 Dbit 7 D 127
n 131 M 130 Dbit 65 D 0
n 133 M 18 Dbit 8 D 1971
n 135 M 36 Dbit 17 D 509033161
n 137 M 68 Dbit 34 D 2154364271382137415
n 139 M 138 Dbit 69 D 0
n 141 M 46 Dbit 23 D 499069107643
n 143 M 60 Dbit 25 D 8062388144103825
n 145 M 28 Dbit 14 D 1851279
n 147 M 42 Dbit 20 D 29918683749
n 149 M 148 Dbit 74 D 0
n 151 M 15 Dbit 5 D 217
n 153 M 24 Dbit 10 D 109655
n 155 M 20 Dbit 8 D 6765
n 157 M 52 Dbit 26 D 28685347945035
n 159 M 52 Dbit 21 D 28324525958305
n 161 M 33 Dbit 15 D 53353631
n 163 M 162 Dbit 81 D 0
n 165 M 20 Dbit 7 D 6355
n 167 M 83 Dbit 36 D 0
n 169 M 156 Dbit 78 D 0
n 171 M 18 Dbit 9 D 1533
n 173 M 172 Dbit 86 D 0
n 175 M 60 Dbit 29 D 6588122883467697
n 177 M 58 Dbit 29 D 1628420204246959
n 179 M 178 Dbit 89 D 0
n 181 M 180 Dbit 90 D 0
n 183 M 60 Dbit 26 D 6300117511512825
n 185 M 36 Dbit 18 D 371456631
n 187 M 40 Dbit 21 D 5879741325
n 189 M 18 Dbit 7 D 1387
n 191 M 95 Dbit 41 D 0
n 193 M 96 Dbit 48 D 0
n 195 M 12 Dbit 3 D 21
n 197 M 196 Dbit 98 D 0
n 199 M 99 Dbit 45 D 0
n 201 M 66 Dbit 33 D 367099384551433863
n 203 M 84 Dbit 42 D 0
n 205 M 20 Dbit 10 D 5115
n 207 M 66 Dbit 33 D 356458822680377809
n 209 M 90 Dbit 45 D 0
n 211 M 210 Dbit 105 D 0
n 213 M 70 Dbit 35 D 5542683665339959171
n 215 M 28 Dbit 9 D 1248537
n 217 M 15 Dbit 5 D 151
n 219 M 18 Dbit 6 D 1197
n 221 M 24 Dbit 7 D 75915
n 223 M 37 Dbit 13 D 616318177
n 225 M 60 Dbit 29 D 5124095576030431
n 227 M 226 Dbit 113 D 0
n 229 M 76 Dbit 38 D 0
n 231 M 30 Dbit 12 D 4648233
n 233 M 29 Dbit 10 D 2304167
n 235 M 92 Dbit 41 D 0
n 237 M 78 Dbit 34 D 0
n 239 M 119 Dbit 52 D 0
n 241 M 24 Dbit 12 D 69615
n 243 M 162 Dbit 81 D 0
n 245 M 84 Dbit 41 D 0
n 247 M 36 Dbit 15 D 278216505
n 249 M 82 Dbit 41 D 0
n 251 M 50 Dbit 25 D 4485656999373
n 253 M 110 Dbit 52 D 0
n 255 M 8 Dbit 1 D 1
n 257 M 16 Dbit 8 D 255
n 259 M 36 Dbit 15 D 265326165
n 261 M 84 Dbit 39 D 0
n 263 M 131 Dbit 59 D 0
n 265 M 52 Dbit 26 D 16994715574983
n 267 M 22 Dbit 10 D 15709
n 269 M 268 Dbit 134 D 0
n 271 M 135 Dbit 62 D 0
n 273 M 12 Dbit 4 D 15
n 275 M 20 Dbit 8 D 3813
n 277 M 92 Dbit 46 D 0
n 279 M 30 Dbit 13 D 3848537
n 281 M 70 Dbit 35 D 4201393668033492183
n 283 M 94 Dbit 47 D 0
n 285 M 36 Dbit 18 D 241120971
n 287 M 60 Dbit 26 D 4017148099675425
n 289 M 136 Dbit 68 D 0
n 291 M 48 Dbit 21 D 967267961205
n 293 M 292 Dbit 146 D 0
n 295 M 116 Dbit 54 D 0
n 297 M 90 Dbit 45 D 0
n 299 M 132 Dbit 66 D 0
n 301 M 42 Dbit 18 D 14611450203
n 303 M 100 Dbit 45 D 0
n 305 M 60 Dbit 30 D 3780070506907695
n 307 M 102 Dbit 51 D 0
n 309 M 102 Dbit 46 D 0
n 311 M 155 Dbit 68 D 0
n 313 M 156 Dbit 78 D 0
n 315 M 12 Dbit 3 D 13
n 317 M 316 Dbit 158 D 0
n 319 M 140 Dbit 65 D 0
n 321 M 106 Dbit 53 D 0
n 323 M 72 Dbit 37 D 0
n 325 M 60 Dbit 30 D 3547450783405683
n 327 M 36 Dbit 13 D 210151305
n 329 M 69 Dbit 34 D 1794212189540138759
n 331 M 30 Dbit 15 D 3243933
n 333 M 36 Dbit 14 D 206364795
n 335 M 132 Dbit 57 D 0
n 337 M 21 Dbit 7 D 6223
n 339 M 28 Dbit 8 D 791845
n 341 M 10 Dbit 2 D 3
n 343 M 147 Dbit 73 D 0
n 345 M 44 Dbit 22 D 50991843607
n 347 M 346 Dbit 173 D 0
n 349 M 348 Dbit 174 D 0
n 351 M 36 Dbit 15 D 195781985
n 353 M 88 Dbit 44 D 0
n 355 M 140 Dbit 70 D 0
n 357 M 24 Dbit 10 D 46995
n 359 M 179 Dbit 80 D 0
n 361 M 342 Dbit 171 D 0
n 363 M 110 Dbit 55 D 0
n 365 M 36 Dbit 17 D 188272539
n 367 M 183 Dbit 87 D 0
n 369 M 60 Dbit 26 D 3124448521969775
n 371 M 156 Dbit 78 D 0
n 373 M 372 Dbit 186 D 0
n 375 M 100 Dbit 49 D 0
n 377 M 84 Dbit 42 D 0
n 379 M 378 Dbit 189 D 0
n 381 M 14 Dbit 4 D 43
n 383 M 191 Dbit 87 D 0
n 385 M 60 Dbit 30 D 2994601310667135
n 387 M 42 Dbit 21 D 11364461269
n 389 M 388 Dbit 194 D 0
n 391 M 88 Dbit 39 D 0
n 393 M 130 Dbit 65 D 0
n 395 M 156 Dbit 78 D 0
n 397 M 44 Dbit 22 D 44312811195
n 399 M 18 Dbit 4 D 657

[carpetpool]

Originally Posted by Dr Sardonicus

...

For example, 346 is the multiplicative order of 2 (mod 1093) and also (mod 193^2).

2^346 = 12486 (mod 193^2), not 1 (mod 193^2).

Too late to fix the typo?