It is well known that
eπ√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
x2 – 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
x2 – 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.
eπ√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
x2 – 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,
x2 – 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.
eπ√43 = 884,736,743.999777… which likewise is close to an integer.
This in turn is closely related to solutions for the quadratic expression,
x2 – 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
x2 – 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.
eπ√17 = 442,150.99767568… which again suggests an integer result provides an interesting case.
This is closely related with the quadratic equation
x2 – 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
x2 – 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
x2 – x + 5 = 0 are (1 + √19 i)/2 and (1 – √19 i)/2.
Now eπ√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
x2 – x + 35 = 0.
Though this expression does not generate successive primes it does appear to generate a large number of them!