541456 and 51456. I checked 20 numbers 2000 times and found 200 patterns!!

[Dr Sardonicus]

Originally Posted by enzocreti

the probable primes pg(51456) and pg(541456) end both with digits 55967! by the way 55967 is prime

So, you've got (2^k1 - 1)*10^m1 + 2^*(k1 - 1) - 1 and (2^k2 - 1)*10^m2 + 2^*(k2 - 1) - 1.

As long as m1 and m2 are at least 5, (which they are here) the last five decimal digits of your pg numbers agree with those of 2^*(k1 - 1) - 1 and 2^*(k2 - 1) - 1, respectively. The last five digits of these two numbers are the same if their difference is divisible by 10^5.

Assuming k1 < k2, subtraction gives

2^(k1 - 1)*(2^(k2 - k1) - 1).

Now if k1 - 1 is at least 5 (which it is here), the difference of your two pg numbers is automatically divisible by 2^5. Check. And if k2 - k2 is divisible by 4*5^4, the difference will also be divisible by 5^5. Let's see. k2 - k1 = 490000, which is divisible by 4*5^4. Double-check. The difference is divisible by 10^5.

So it is absolutely no surprise that the two numbers have the same last five decimal digits.

[enzocreti]

pg(541456) and p(51456) are congruent to 46 mod 67, any explanation?

[Batalov]

My phone number is congruent to 46 mod 201, any explanation?

[enzocreti]

pg(51456) and pg(541456) have the same residue mod 67...strange

[science_man_88]

Originally Posted by enzocreti

pg(51456) and pg(541456) have the same residue mod 67...strange

two numbers have the same residue mod all divisors of their difference.

[enzocreti]

Originally Posted by science_man_88

two numbers have the same residue mod all divisors of their difference.

so pg(541456)-pg(51456) is a multiple of 67?

[enzocreti]

Is it true that:


51456 divides

pg(541456)-pg(51456)?

[science_man_88]

Originally Posted by enzocreti

so pg(541456)-pg(51456) is a multiple of 67?

yes that's all congruence in modular arithmetic is at the most basic level. here's something else to help you:

https://en.m.wikipedia.org/wiki/Pigeonhole_principle

[Dr Sardonicus]

Originally Posted by enzocreti

pg(51456) and pg(541456) have the same residue mod 67...strange

They have the same residue mod 2^15490, 3^2, 5^5, 17, 41, 67, and 257. But not modulo any other prime less than 50 million. Explain! And, all but three of these primes are Fermat primes. Explain!

[enzocreti]

i cannot explain nothing

[Dr Sardonicus]

Originally Posted by enzocreti

i cannot explain nothing

Perhaps there is nothing to explain.