|
To All,
Thank you for your comments. Below is an
updated and neater attempt to prove that P =
2^170141183460469231731687303715884105727 - 1 is Prime Number
Regards,
Leo de Velez
Updated Proof for Biggest Prime Number
P = 2^170141183460469231731687303715884105727 - 1 is Prime Number
PROOF: Given: 2^127 -1 = 170141183460469231731687303715884105727 is prime number. P = 2^170141183460469231731687303715884105727 - 1 is a number with 170141183460469231731687303715884105727 binary digits and all binary digits are equal to 1
For prime number q < 170141183460469231731687303715884105727 2^(q-1) mod q = 1 ==comment== (q-1) is the number of integers less than q that is not a factor of q plus the number 1 Then, 2^(q-1) -1 = x * q where x is any positive integer If N = 2^(q-1) - 1 = x * q then, N is a number with all binary digits equal to 1 and the number of digits is EVEN (q-1). and N is the LOWEST VALUE of MULTIPLE of q with all binary digits equal to 1 (very important!!!) The next multiple of q with all binary digits equal to 1 is M = 2^(2*(q-1)) - 1 and its number of binary digits is equal to 2*(q-1) and so on..... Since 170141183460469231731687303715884105727 is a prime number, it is not a multiple of (q-1) So the 170141183460469231731687303715884105727 binary digits number (prime number digits) with all binary digits equal to 1 cannot be exactly be divided by the number N with x*(q-1) binary digits with all binary digits equal to 1. THEREFORE, P is NOT a multiple of q where q is prime number less than 170141183460469231731687303715884105727
For prime number q > 170141183460469231731687303715884105727 the least value of multiple of q with all binary digits equal to 1 is N = 2^(q-1) -1 = x * q Since N is greater than P for q > 170141183460469231731687303715884105727 THEREFORE, P is NOT a multiple of q where q is a prime number greater than 170141183460469231731687303715884105727
Since there are NO prime number q <> 170141183460469231731687303715884105727 that has a multiple N that has binary digits all equal to 1 that can divide P THEREFORE, P = 2^170141183460469231731687303715884105727 - 1 is Prime Number
OPTIONAL - (BUT ALREADY COVERED BY TWO GROUPS OF Q ABOVE) For a prime number q with ALL binary digits NOT equal to 1, and N is a number with all binary digits equal to 1, N is THE least multiple of prime number q if N = 2^(q-1) - 1. It has (q-1) binary digits all equal to 1. The next multiple of q with binary digits all equal to 1 has number of binary digits equal to 2*(q-1) and so on..... Since 170141183460469231731687303715884105727 is a prime number, it is not a multiple of (q-1) So the 170141183460469231731687303715884105727 binary digits number (prime number digits) with all binary digits equal to 1 cannot be exactly be divided by the number N with x*(q-1) binary digits with all binary digits equal to 1. THEREFORE, P is NOT a multiple of prime number q AND THEREFORE P = 2^170141183460469231731687303715884105727 - 1 is Prime Number
|
