It is well
known that

e

^{π√163}^{ }comes very close to being a whole integer.
In fact it
results in the number 262, 537, 412, 640, 768,743.999999999999.25 (approx)
which would be highly unlikely to occur on a random basis.

In fact it
is not accidental with the appearance of √163 directly related to the solutions
of the quadratic equation

x

^{2}– x + 41 = 0, i.e. (1 + √163 i)/2 and (1 – √163 i)/2.
And as
Euler showed, a special significance attaches to this quadratic equation in
generating successive prime numbers.

So, for all
natural number values of x from 1 to 40

x

^{2}– x + 41 results in a prime number.
This
automatic generation of prime numbers breaks down with x = 41.

Though the
automatic pattern is therefore broken, the expression will still generate a
high proportion of primes from x > 41.

For example
for values of x from 42 to 50 (inclusive) the expression will generate prime
numbers for x = 43, 44, 46, 47, 48 and 49.

However, there are
several other expressions for x < 41 where a similar pattern is in evidence.

For example,

e

^{π√67}^{ }= 147,107,952,743.99999866… which again is very close to an integer result.
This in
turn is closely related to solutions for the quadratic expression

x

^{2}– x + 17 = 0, i.e. (1 + √67 i)/2 and (1 – √67 i)/2.
And once
more for all values of x, this time from 1 to 16 inclusive,

x

^{2}– x + 17 generates successive prime numbers.
Again
though the automatic pattern breaks down for x = 17, the expression still
remains a remarkably good generator of prime numbers.

So for
example from x = 18 to x = 27 (inclusive) primes are generated in all cases
except for x = 21 and x = 26.

Again,

e

^{π√43}^{ }= 884,736,743.999777… which likewise is close to an integer.
This in
turn is closely related to solutions for the quadratic expression,

x

^{2}– x + 11 = 0, i.e. (1 + √43 i)/2 and (1 – √43 i)/2.
And again
for all values of x from 1 to 10 inclusive

x

^{2}– x + 11 generates successive prime numbers.
Though the
automatic pattern breaks down with x = 11, the expression still remains a good
generator of primes.

From x = 12
to x = 21 (inclusive), primes are generated for x = 13, 14, 16, 17, 18, 19 and
21.

e

^{π√17}^{ }= 442,150.99767568… which again suggests an integer result provides an interesting case.
This is
closely related with the quadratic equation

x

^{2}– x + 4.5 = 0, i.e. (1 + √17 i)/2 and (1 – √17 i)/2.
Now the
problem here is that 4.5 is not an integer.

However
when rounded upwards to 5

x

^{2}– x + 5 generates successive primes from 1 to 4 inclusive.
However
this represents success over a very small range where the frequency of primes
in any case is especially high.

Also, the
expression does not serve as a very good predictor of primes from outside this
range.

There are other interesting patterns that are perhaps in evidence.

The 3 numbers that unambiguously fulfill the same patterns of prime generation are related to 43, 67 and 163 respectively. Now the differences between these numbers are 24 and 96 (of which 24 is a factor).

Interestingly 19 likewise differs from 43 by 24 and this number naturally leads to 5 in the corresponding quadratic expression.

In other words the solutions for

x

^{2}– x + 5 = 0 are (1 + √19 i)/2 and (1 – √19 i)/2.
Now e

^{π√19}^{ }= 885,449.77768....
This does not approximate an integer but is however close to a number ending with the fraction 7/9!

Likewise 139 = 67 + (24*3) is a prime number and directly associated with the quadratic equation

x

^{2}– x + 35 = 0.
Though this expression does not generate successive primes it does appear to generate a large number of them!

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