## Primes and Composites occurring in Certain Equations

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• Conjectures concerning Primes and Composites of the Form 20x^2 -1, 20x^2 + 20x + 1, 20x^2 + 20x - 11, 20x^2 + 60x - 19, 20x^2 +180x +149 and 20x^2 + 340x +
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Conjectures concerning Primes and Composites of the Form
20x^2 -1, 20x^2 + 20x + 1, 20x^2 + 20x - 11,
20x^2 + 60x - 19, 20x^2 +180x +149 and 20x^2 + 340x + 421:.

These equations are the first six members of a family of
forms derived originally from 5x^2 + 5x +1
(from the puzzle of 9/8/8), and they share some of its
special properties. Prime and composite terms may be
distinguished by simply counting the number of squares
ocurring in in a relatively small calculated interval. Although
this assertion remains to be proven, it seems a good bet,
as the first 10000 iterations of each sequence yielded no
counterexamples. All primes composing terms of each of
these six equations are always +/- 1 mod 10 , and the
constant second difference between three consecutive
terms is always 40.

The first two conjectures are a recasting of 5) and 6)
from the puzzle of 12/8/8 (still underway). This recasting
will now include in the count of squares the two non-
functional squares occuring just outside of the interval
that we examined then (old interval + 2) .In all cases a
count of squares equal to two indicates a term is prime,
and three or more that it is composite. This way of
knowing extends to all six equations, and will also extend
to additional members of the family (infinite?) of forms
that can be derived.

For All Conjectures below:
x, A(x), k, T(k) : integers;
T(k) = 5*(2*k -1)^2 - 4*A(x) ;

Let A(x) = 20x^2 - 1;
5a) A(x) will be prime if exactly two k values
exist in the interval 2x < k < 3x + 2 such that T(k) is a
square of an integer.

Example: if x = 5, then A(x) = 499, 10< k < 17
and the values of T(k) are:
209, 649, 1129, 1649 , 47^2, 53^2.
Two of these are squares, therefore 499 is prime.

Let A(x) = 20x^2 - 1;
6a) A(x) will be composite if there exists at
least three k values in the interval 2x < k < 3x + 2
such that T(k) is a square of an integer.

Example: if x = 4, then A(x) = 319, 8< k < 14
and the values of T(k) are:
13^2, 23^2, 929 ,37^2, 43^2.
Four of these are squares, therefore 319 is composite.

Let A(x) = 20x^2 + 20x + 1;
9) A(x) will be prime if exactly two k values
exist in the interval 2x +1 < k < 3x + 4 such
that T(k) is a square of an integer.

Example: if x = 3, then A(x) = 241, 7< k < 13
and the values of T(k) are:
161, 481, 29^2, 1241, 41^2
Two of these are squares, therefore 241 is prime.

Let A(x) = 20x^2 + 20x + 1;
10) A(x) will be composite if there exists at
least three k values in the interval 2x + 1< k < 3x + 4
such that T(k) is a square of an integer.

Example: if x = 2, then A(x) = 121, 5< k < 10
and the values of T(k) are:
11^, 19^2 ,641, 31^2.
Three of these are squares, therefore 121 is composite.

Let A(x) = 20x^2 + 20x - 11;
11) A(x) will be prime if exactly two k values
exist in the interval 2x + 1 < k < 3x + 5 such
that T(k) is a square of an integer.

Example: if x = 4, then A(x) = 389, 9< k < 17
and the values of T(k) are:
249, 649, 33^2, 1569, 2089, 2649, 57^2
Two of these are squares, therefore 389 is prime.

Let A(x) = 20x^2 + 20x - 11;
12) A(x) will be composite if there exists at
least three k values in the interval 2x + 1< k < 3x + 5
such that T(k) is a square of an integer.

Example: if x = 5, then A(x) = 589, 11< k < 20
and the values of T(k) are:
17^2, 769, 1289, 43^2, 2449, 3089, 3769, 67^2
Three of these are squares, therefore 589 is composite

Let A(x) = 20x^2 + 60x - 19;
13) A(x) will be prime if exactly two k values
exist in the interval 2x +3 < k < 3x + 10
such that T(k) is a square of an integer.

(*note: from this point on only the square values
of T(k), and the first value of T(k) will appear in
the examples. All other T(k) will be designated as '-'.

Example: if x = 4, then A(x) = 541, 11< k < 22 ,
and the values of T(k) are:
481 31^2 - - - - - - - 79^2
Two of these are squares, therefore 541 is prime.

Let A(x) = 20x^2 + 60x - 19;
14) A(x) will be composite if there exists at
least three k values in the interval 2x +3< k < 3x + 10
such that T(k) is a square of an integer.

Example: if x = 3, then A(x) = 341, 9< k < 19
and the values of T(k) are:
21^2 29^2 - - - - - - 69^2
Three of these are squares, therefore 341 is composite.

Let A(x) = 20x^2 +180x +149;
15) A(x) will be prime if exactly two k values
exist in the interval 2x + 9< k < 3x + 23
such that T(k) is a square of an integer.

Example: if x = 1, then A(x) = 349, 11< k < 26
and the values of T(k) are:
1249 -- 53^2 - - - - - - - - - 103^2
Two of these are squares, therefore 349 is prime.

Let A(x) = 20x^2 +180x +149;
16) A(x) will be composite if there exists at
least three k values in the interval
2x + 9< k < 3x + 23 such that T(k) is a
square of an integer.

Example: if x = 2, then A(x) = 589, 13< k < 29
and the values of T(k) are:
1289 43^2 - - - 67^2 - - 83^2 - - - - - 113^2
Four of these are squares, therefore 589 is composite.

Let A(x) = 20x^2 + 340x + 421;
17) A(x) will be prime if exactly two k values
exist in the interval 2x + 17< k < 3x + 43
such that T(k) is a square of an integer.

Example: if x = 5, then A(x) = 2621, 27< k < 58
and the values of T(k) are:
4641 - - - - - - - - - - - - - - - - - - 181^2
- - - - - - - - - 231^2
Two of these are squares, therefore 2621 is prime.

Let A(x) = 20x^2 + 340x + 421;
18) A(x) will be composite if there exists at
least three k values in the interval
2x + 17 < k < 3x + 43 such that T(k) is a
square of an integer.

Example: if x = 4, then A(x) = 2101, 25< k < 55
and the values of T(k) are:
4601 - - - - 101^2 - - - - - - - -151^2
- - - - - - - - - - - - - 221^2
Three of these are squares, therefore 2101 is composite.

Aldrich Stevens
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