Using Logarithms of Mersenne Primes

[hydeer]

Wolfram mathematica is used to make a list of Log, to the base 2 ,of Mersenne primes .

t[n_] := N[ Log2[2^Prime[n] - 1]]
Table[t[i], {i, 1, 50}]
{1.58496, 2.80735, 4.9542, 6.98868, 10.9993, 12.9998, 17., 19., 23., \
29., 31., 37., 41., 43., 47., 53., 59., 61., 67., 71., 73., 79., 83., \
89., 97., 101., 103., 107., 109., 113., 127., 131., 137., 139., 149., \
151., 157., 163., 167., 173., 179., 181., 191., 193., 197., 199., \
211., 223., 227., 229.}

[science_man_88]

Originally Posted by hydeer

Wolfram mathematica is used to make a list of Log, to the base 2 ,of Mersenne primes .

t[n_] := N[ Log2[2^Prime[n] - 1]]
Table[t[i], {i, 1, 50}]
{1.58496, 2.80735, 4.9542, 6.98868, 10.9993, 12.9998, 17., 19., 23., \
29., 31., 37., 41., 43., 47., 53., 59., 61., 67., 71., 73., 79., 83., \
89., 97., 101., 103., 107., 109., 113., 127., 131., 137., 139., 149., \
151., 157., 163., 167., 173., 179., 181., 191., 193., 197., 199., \
211., 223., 227., 229.}

And this proves ?

[a1call]

Originally Posted by hydeer

Wolfram mathematica is used to make a list of Log, to the base 2 ,of Mersenne primes .

t[n_] := N[ Log2[2^Prime[n] - 1]]
Table[t[i], {i, 1, 50}]

I think you mean Merseene-Numbers with Prime-Exponents, not Mersenne-Primes.

Regardless, I think that's a very interesting if not useful observation/discovery.

ETA then again you are getting the same prime as the exponent after rounding the log base 2 of the Mersenne number. So not interesting anymore.

[a1call]

Here is the PARI GP equivalent with rounding added:

forprime(p=2,19^2,{
  M= 2^p-1;
    theRoundedLog =round(log(M)/log(2));
    print("*** M",p," >>> ", theRoundedLog ," >> ",isprime(theRoundedLog ));
    if(!isprime(theRoundedLog ),next(19););
  
})

*** M2 >>> 2 >> 1
*** M3 >>> 3 >> 1
*** M5 >>> 5 >> 1
*** M7 >>> 7 >> 1
*** M11 >>> 11 >> 1
*** M13 >>> 13 >> 1
*** M17 >>> 17 >> 1
*** M19 >>> 19 >> 1
*** M23 >>> 23 >> 1
*** M29 >>> 29 >> 1
*** M31 >>> 31 >> 1
*** M37 >>> 37 >> 1
*** M41 >>> 41 >> 1
*** M43 >>> 43 >> 1
*** M47 >>> 47 >> 1
*** M53 >>> 53 >> 1
*** M59 >>> 59 >> 1
*** M61 >>> 61 >> 1
*** M67 >>> 67 >> 1
*** M71 >>> 71 >> 1
*** M73 >>> 73 >> 1
*** M79 >>> 79 >> 1
*** M83 >>> 83 >> 1
*** M89 >>> 89 >> 1
*** M97 >>> 97 >> 1
*** M101 >>> 101 >> 1
*** M103 >>> 103 >> 1
*** M107 >>> 107 >> 1
*** M109 >>> 109 >> 1
*** M113 >>> 113 >> 1
*** M127 >>> 127 >> 1
*** M131 >>> 131 >> 1
*** M137 >>> 137 >> 1
*** M139 >>> 139 >> 1
*** M149 >>> 149 >> 1
*** M151 >>> 151 >> 1
*** M157 >>> 157 >> 1
*** M163 >>> 163 >> 1
*** M167 >>> 167 >> 1
*** M173 >>> 173 >> 1
*** M179 >>> 179 >> 1
*** M181 >>> 181 >> 1
*** M191 >>> 191 >> 1
*** M193 >>> 193 >> 1
*** M197 >>> 197 >> 1
*** M199 >>> 199 >> 1
*** M211 >>> 211 >> 1
*** M223 >>> 223 >> 1
*** M227 >>> 227 >> 1
*** M229 >>> 229 >> 1
*** M233 >>> 233 >> 1
*** M239 >>> 239 >> 1
*** M241 >>> 241 >> 1
*** M251 >>> 251 >> 1
*** M257 >>> 257 >> 1
*** M263 >>> 263 >> 1
*** M269 >>> 269 >> 1
*** M271 >>> 271 >> 1
*** M277 >>> 277 >> 1
*** M281 >>> 281 >> 1
*** M283 >>> 283 >> 1
*** M293 >>> 293 >> 1
*** M307 >>> 307 >> 1
*** M311 >>> 311 >> 1
*** M313 >>> 313 >> 1
*** M317 >>> 317 >> 1
*** M331 >>> 331 >> 1
*** M337 >>> 337 >> 1
*** M347 >>> 347 >> 1
*** M349 >>> 349 >> 1
*** M353 >>> 353 >> 1
*** M359 >>> 359 >> 1

ETA you would get exact match with no rounding required if you remove the "-1" from your code.

[Dr Sardonicus]

Originally Posted by hydeer

Wolfram mathematica is used to make a list of Log, to the base 2 ,of Mersenne primes .
[snip]

Whether 2^p - 1 is prime or not, if p is at all large, the base-two log of 2^p - 1 should be fairly close to

p - 1/(2^(p)*ln(2).)

Of course,

p - 1/(2^(p)*ln(2)) - 1/(2^(2*p+1)*ln(2))

would be closer...