same residue
 

the two probable primes pg(541456) and pg(51456) have the same residue mod 67: 46

[QUOTE=enzocreti;503250]the two probable primes pg(541456) and pg(51456) have the same residue mod 67: 46[/QUOTE]

again all this implies is there difference is divisible by 67.

yes
 

[QUOTE=science_man_88;503251]again all this implies is there difference is divisible by 67.[/QUOTE]

yes the two probable primes have also the same residue mod 201

[QUOTE=enzocreti;503253]yes the two probable primes have also the same residue mod 201[/QUOTE]

that just means they are different by a multiple of 201= 3*67

residue
 

the two probable primes have also the same residue mod 511

[QUOTE=enzocreti;503258]the two probable primes have also the same residue mod 511[/QUOTE]

that just means they have a difference divisible by 511=7*73; together with our 201 fact we get that the difference is also divisible by lcm(201,511)= lcm(3*67,7*73)=3*67*7*73=102711

residue
 

the two probables primes have the same residue 2^n modulo Fermat primes 257 and 17

[QUOTE=enzocreti;503260]the two probables primes have the same residue 2^n modulo Fermat primes 257 and 17[/QUOTE]

adding to our list of prime factors, we get it is divisible by lcm(102711,257,17) = 102711*257*17=448744359

[QUOTE=enzocreti;503260]the two probables primes have the same residue 2^n modulo Fermat primes 257 and 17[/QUOTE]
Take two random numbers, a and b.

Next, for m from 2 to 100000, check if a and b have the same modulo value. You will almost always find several m's.

Stop obsessing about it. It is not even a coincidence - it is a multiple testing result. Compare it to this: take two random people, then ask them 100,000 different questions (what date you were born on, what was the day of the week, what province you are from, ...etc) and yell in excitement every time their answers are the same. It is painfully obvious that if you ask [B]one [/B]question and the answers are the same then that could be surprising, but if you ask [B]thousands of times[/B] then the matching answers will not be surprising at all.

Did you read the [URL="https://www.mersenneforum.org/showthread.php?p=502910#post502910"]Wikipedia article that you were given[/URL]? If you didn't and you continue 'spamming' (your posts constitute spam -- in case you didn't know it), you may lose your posting privileges.

[QUOTE=enzocreti;503244]pg(51456) and pg(541456) are probable primes.


Why their difference is a multiple of 67?[/QUOTE]

You have 38 probable primes which have among them binomial(38, 2) = 703 differences; you should expect about 10 of them to be multiples of 67 just by chance. As it turns out, 9 of the differences are.

[code]ex=[2,3,4,7,8,12,19,22,36,46,51,67,79,215,359,394,451,1323,2131,3336,3371,6231,19179,39699,51456,56238,69660,75894,79798,92020,174968, 176006,181015,285019,331259,360787,366770,541456];
glue(x,y)=10^#Str(y)*x+y
pg(k)=glue(2^k-1,2^(k-1)-1)
pgm(k,m)=if(k<999,return(Mod(pg(k),m))); my(d=(k-1)*log(2)\log(10)+1, n=Mod(2, m)^(k-1)); (2*n-1)*Mod(10,m)^d + n-1;
addhelp(pg, "pg(k): Returns enzocreti's pg(k), the decimal concatenation of the k-th and (k-1)-th Mersenne numbers.");
addhelp(pgm, "pgm(k,m): Returns pg(k) mod m.");

sum(i=2,#ex,sum(j=1,i-1,pgm(ex[j],67)==pgm(ex[i],67)))[/code]

The [url=http://blog.minitab.com/blog/adventures-in-statistics-2/how-to-correctly-interpret-p-values]p-value[/url] is 0.7217. I'm not impressed.