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 = 3is a prime number with all binary
digits equal to 1(total of 2 binary digits)2^(2^2 -1) -1 =
2^3 - 1 = 127is a prime number with all binary digits equal to 1(total
of 3 binary digits)2^(2^(2^2 -1) -1) -1 =
170141183460469231731687303715884105727is a prime number with all binary
digits equal to 1(total of 127 binary digits)So it follows
that
2^170141183460469231731687303715884105727 - 1IS
ALSO A PRIME NUMBER WITH ALL BINARY DIGITS EQUAL TO 1(TOTAL OF
170141183460469231731687303715884105727 DIGITS)And so on.
PROOFIf q is any prime
number,then 2^(q-1) mod q = 1and 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) -1orq is a factor of a number N with (q-1) binary
digits all equal to 1This number N has EVEN number of binary digits all
equal to 1P = 2^170141183460469231731687303715884105727 - 1So P
is a numberwith a prime number (170141183460469231731687303715884105727) of
binarydigits 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 digitsall equal to
1.Therefore, from binary division,Prime Number of Binary Digits
All Equal to 1DIVIDED BYEven Number of Binary Digits All Equal to
1HAS A REMAINDERSOany prime numbers q less than
170141183460469231731687303715884105727is NOT a factor of P =
2^170141183460469231731687303715884105727 - 1It also follows from
binary division thatFor ALL numbers k less than P with binary digit all
equal to 1,k is NOT a factor of PJust to remove all EVEN
numbers,ALL even numbers E less than P is not a factor of P.NOW,
THE FINAL ELIMINATIONFor 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) - 1N is greater than
2^170141183460469231731687303715884105727 -1Therefore,All q
170141183460469231731687303715884105727Is NOT a factor of P =
2^170141183460469231731687303715884105727 - 1AND THEREFORE,P
= 2^170141183460469231731687303715884105727 - 1IS A PRIME NUMBER
Regards,
Leo de Velez
26B Prudent Lane,
Sanville Subdivision,
Quezon City, Philippines
+63 917 532 9297
