A proximate-prime polynomial
is simply a quadratic equation - a finite polynomial of
the second degree - that is derived from four successive
(proximate, or neighboring) primes. Proximate-prime
polynomials are interesting because they exhibit much
greater prime densities than other polynomials.

When you graph
primes against an X-axis that treats the expanding
interval between successive perfect squares as a constant
unit subdivided into equal parts, you produce a
distinctive wave form for primes and prime factors.

For every composite number that is
not itself a perfect square there exists a pair of
nonconsecutive perfect squares whose difference is equal
to the composite. Even before we get to the subject of
factorization, the consequences of this observation are
fascinating and far-reaching.

It began with an
exploration of biquadratic paired primes: 2
primes separated by the equivalent of exactly 2
quadratic intervals.... Then the investigation took the
logical next level by asking the question: Are there
prime pairs that are separated by other, greater
multiples of the quadratic interval? And if there are,
what are the frequency characteristics by interval size
and perfect square offset? The results are in, with
charts, an Excel visualization, and masses of
half-digested data...!

Find examples throughout this site that demonstrate using
VBA code with worksheets and graphing - including
generating primes, perfect squares, and composites, doing
modular arithmetic, calculating GCDs, and more....

Desktop program and complete project source code
for implementing the gold standard in primality
testing. A fast and reliable test for numbers up
to 10^{27}-1 (that's 1 with 26 9s - a
prime number...!).
(The project illustrates how to use a legacy
language, VB6, not designed for big integers. It
includes modular exponentiation code by DI Management Cryptography
Software.)

"Fermatic" is a made-up word: Fermat + Automatic.
This tool takes Fermat's great theorem
to the limit, with some experiments to weed out
pesky pseudoprimes. Rapidly generate prime,
pseudoprime, and composite data.

Enter 3 or 4 numbers in a sequence and find out
what the next 10, next 1,000, or next 10,000
values are. QTest lets you derive a quadratic
equation from the values you input (and solve the
equation's roots). Then you can use this
polynomial to generate and analyze long number
sequences for primality. (See Robert Sacks' method
for quadratic derivation, used
in Vortex.)

Calculate the products of infinite series using
almost any inputs you can think of. Generates real
number zeta series, demonstrating the calculation
of many important constants - including e, pi, and
phi, the Basel equality and Apéry's constant.