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Dear All,
Below is my humble attempt to find the biggest
prime number without the luxury of extensive computer power. I hope you
will find this interesting and hopefully some of you can check its
validity.
Regards,
Leo de Velez
26B Prudent Lane,
Sanville Subdivision,
Quezon City, Philippines
+63 917 532 9297
Biggest Prime Number
2^2 -1 = 3
is a prime number with all binary digits equal to 1 (total of 2 binary digits) 2^(2^2 -1) -1 = 2^3 - 1 = 127 is a prime number with all binary digits equal to 1 (total of 3 binary digits) 2^(2^(2^2 -1) -1) -1 = 170141183460469231731687303715884105727 is a prime number with all binary digits equal to 1 (total of 127 binary digits) So it follows that 2^170141183460469231731687303715884105727 - 1
IS ALSO A PRIME NUMBER WITH ALL BINARY DIGITS EQUAL TO 1 (TOTAL OF 170141183460469231731687303715884105727 DIGITS) And so on. PROOF If q is any prime number, then 2^(q-1) mod q = 1 and then 2^(q-1) -1 = a * q, where a is an integer less than 2^(q-1) This means that any prime number q is a factor of N = 2^(q-1) -1 or q is a factor of a number N with (q-1) binary digits all equal to 1 This number N has EVEN number of binary digits all equal to 1 P = 2^170141183460469231731687303715884105727 - 1 So P is a number with a prime number (170141183460469231731687303715884105727) of binary digits all equal to 1. For each prime number q less than 170141183460469231731687303715884105727, q is a factor of a number N = 2^(q-1) with an EVEN number of binary digits all equal to 1. Therefore, from binary division, Prime Number of Binary Digits All Equal to 1 DIVIDED BY Even Number of Binary Digits All Equal to 1 HAS A REMAINDER SO any prime numbers q less than 170141183460469231731687303715884105727 is NOT a factor of P = 2^170141183460469231731687303715884105727 - 1 It also follows from binary division that For ALL numbers k less than P with binary digit all equal to 1, k is NOT a factor of P Just to remove all EVEN numbers, ALL even numbers E less than P is not a factor of P. NOW, THE FINAL ELIMINATION For any prime number q greater than 170141183460469231731687303715884105727, the least value of product N = a * q where N has a binary digits all equal to 1 and N = 2^(q-1) - 1 N is greater than 2^170141183460469231731687303715884105727 -1 Therefore, All q > 170141183460469231731687303715884105727 Is NOT a factor of P = 2^170141183460469231731687303715884105727 - 1 AND THEREFORE, P = 2^170141183460469231731687303715884105727 - 1 IS A PRIME NUMBER !!!! Regards, Leo de Velez
26B Prudent Lane,
Sanville Subdivision,
Quezon City, Philippines
+63 917 532 9297
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