Expect at Most One Billionth of a New Fermat Prime!

@article{Boklan2016ExpectAM,
  title={Expect at Most One Billionth of a New Fermat Prime!},
  author={Kent D. Boklan and John H. Conway},
  journal={The Mathematical Intelligencer},
  year={2016},
  volume={39},
  pages={3-5}
}
We provide compelling evidence that all Fermat primes were already known to Fermat. 
4 Citations
Why Does a Prime p Divide a Fermat Number?
Summary A prime dividing a composite Fermat number is called a Fermat prime divisor. Such a prime p must be congruent to 1 modulo 4, and so, by the Fermat–Girard theorem, there exists integers R and
6m Theorem for Prime numbers
We show that for any $P= 6^{m+1}.N -1 $ is a prime number for any $1 13 $ if and only if , $N \ne i^{m+1}Mod(6i+1) +(6i +1)a $ $ ; i,a \le Z^+ $
BiEntropy, TriEntropy and Primality
TLDR
A significant relationship between BiEntropy and TriEntropy is found such that the twin primes conjecture is true and the BiEntropic prime density is shown to be quadratic with a very small Gaussian distributed error.
Frobenius Groups with Perfect Order Classes
The purpose of this paper is to investigate the finite Frobenius groups with “perfect order classes”; that is, those for which the number of elements of each order is a divisor of the order of the

References

SHOWING 1-6 OF 6 REFERENCES
On the second moment for primes in an arithmetic progression
Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic
On the constant in the Mertens product for arithmetic progressions. II: Numerical values
We give explicit numerical values with 100 decimal digits for the constant in the Mertens product over primes in the arithmetic progressions $a \bmod q$, for $q \in \{3$, ..., $100\}$ and $(a, q) =
Factors of generalized Fermat numbers
Generalized Fermat numbers have the form F b,m = b 2m + 1. Their odd prime factors are of the form k . 2 n + 1, k odd, n > m. It is shown that each prime is a factor of some F b,m for approximately
The twenty-fourth Fermat number is composite
TLDR
The rigorous Pepin primality test was performed using independently developed programs running simultaneously on two different, physically separated processors, and it was shown by machine proof that F24 = 2224 + 1 is composite.
An Introduction to the Theory of Numbers
This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,
Expect at most one billionth of a new Fermat Prime
  • 2016