Empty Seat in Mathematical Hall of Fame
In his book Numbers & Such (1968), A.N. Feldzamen claims: "There is a vacant place in the mathematical hall of fame for anyone who can produce a formula that will generate only primes and an infinite number of different primes." (p. 194)
Now, 45 years later, the seat remains vacant. Yet, over the past 400 years, some great mathematicians have tried to take this seat:
So, does a prime-generating formula exist? Will anyone ever take that "vacant place in the mathematical hall fo fame"?
- In 1772, Euler suggested the polynomial n2+n+41, which produces distinct primes for consecutive integers 0 to 39....but fails for n = 41.
- Hardy and Wright extended Euler's formula via (n-40)2+(n-40)+41 = n2-79n+1601, which produces primes for 80 consecutive integers...but...
- In 1640, Fermat offered the formula np = 22p+1 for p a natural number. Now, n1 = 5, n2 = 5, n3 = 257, n4 = 65,537...but it took 100 years until Euler showed that n5 was not a prime number (being divisible by 741)...and it produces composites for p = 6, 7, 8, 9, 11, 12, 18, 23, 36, 38, 73...
- Starting with the unusual constant k = 1.3063..., the formula f(n) = floor(k3n) produces primes for all positive integers, such as f(1) = 2, f(2) = 11, f(3) = 1361, f(4) = 2521008887, ... BUT not all the primes!
- In 1991, Ribenboim found a polynomial in ten variables with integer coefficients such that the set of prime numbers equals the set of positive values of this polynomial when the variables take on all the nonnegative integers
- In 2000, Jones, Sato, Wada, and Wiens found a polynomial of degree 25 in 26 variables, such that its set of positive values are exactly the set of prime numbers...but...
BEWARE! Legendre showed that there is no rational algebraic function that will always produce primes. And, in 1752, Goldbach showed that no polynomial with integer coefficients can give a prime for all integer values.
So, it must be a different type of mathematical animal, such as a formula using weird constants, etc,. Search the Internet and you will find many efforts...Sierpinski, Hardy & Wright, Reimann's Zeta Function...? Let the hunt continue!