and issquare functions:

The primemine programs that were posted at this

PN website needed no reference to issquare functions

to identify which integers occurring as values on

their binary quadratic equations were prime and which

composite; primes and squares of primes GCD(A,y) = 1

appear only once as values of A and all its composites

GCD(A,y) = 1 appear more than once. Nonetheless

the primary equations of both programs had an underlying

unique issquare function bound to them which could if

desired be applied to all of their composites for the

purpose of factoring. In general, it appears that the

existence of such a property in binary quadratics is

quite unusual. This one-to-one correspondence between

issquare and each term seems likely to be a necessary

condition for a binary quadratic form to possess if it

is to be of any use at all in a primality test/factoring

method. For A = 5*x*x + 5*x*y + y*y the accompanying

issquare function is 5r*r -4*A = s*s, and for A = 25*x*x +

25*x*y + y*y it is 21r*r -5*A = s*s.

Finding the correct issquare function is not a trivial

problem to solve for many of the eligible equations.

The issquare function 5r*r -4*A = s*s mentioned above was

probably the easiest one to discover, is unusually easy to

apply, and is bound to what is perhaps the most well-

ordered binary quadratic of them all. It may be observed

that predictable patterns of many values of the issquare

function exist and may be calculated from a simple a

transformation of the x, y coordinates of A (ex: if x = 7,

y = 6, A = 491 then r1 = 2*7 + 6, r2 = 3*7 + 6 and thus

5*20*20 - 4*491 = 36 and 5*27*27 - 4*491 = 1681). If 491

were composite it would exist in at least one more

location among the values of A, and from the two sets of

issquare values found factors could easily be calculated.

Since it is prime it appears only once and indeed because

it appearsonly once it is prime. Hard proof for the

behaviors of issquare functions with their paired binary

quadratic forms are few and far between; a study of them

is more akin to the science of horticulture than to

deductive mathematics. One pattern shades into the next,

properties appear and disappear,and exceptions proliferate

in some kind of uneven continuum from order to chaos.

Tinkering with a few rules, and stretching some assumptions

grows some interesting patterns on the computer screen.

In many binary quadratic forms there are one or two

small factors that are not factors of the y variable

that make some values of A that occur only once look

composite, a serious breach of our basic rule of

primality. Since these 'extraneous' factors are few

and easily handled, however, and the forms in question

retain the other essential properties, it is logical to

tolerate an easing of the rule structure and simply

classify them as unimportant exceptions. It turns out

that there are quite a few equations with this property

that can be considered for inclusion into our previous

list that was expressed as the general equation A =

N*x*x + N*x*y + y*y. That list now includes N = 2,3,5,7,

8,9,10,11,12,13,14,15,16,18,19,20,21,23,24,25,26,33,34,35,

39,40,44,46,48,51,55,56,69,70,74,84,88,91,95,114,124,140,

156,165,168,and 264. In addition another general equation

with a slightly different rules set also looks possibly

useful in primality test/factoring methods:

A = 4*N*(x*x -x + x*y) - 2*N*y + 2*y*y + N for N = 3,4,

5,7,9,11,15,17,23,35,37, and 57. There is no doubt that

others exist and will eventually discovered.

The binary quadratic form of primemine2 (A = 25*x*x + 25*x*y

+ y*y ) is an example of an equation with two ignorable

'extraneous' factors (3 and 7) distributed randomly among

the A values. These however turned out to be somewhat helpful

in ascertaining the issquare function bound to it

(21r*r -5*A = s*s). The application of this function to

factoring composites of A is far more complex and problematic

than was the case in pm1 (**for use as a factoring

method both pm1 and pm2 would need to be adjusted to record

all x,y values and also add new procedures) and it is not

even certain that a way to make it work exists in every

instance. In pm1 finding two locations for a value of A

would make the finding of factors automatic . In pm2 one of

four different patterns may need to be used to calculate the necessary issquare values for factoring depending upon whether

the x,y values for each location of A are odd-odd,odd-even,

even-odd, or even-even. A small grid of issquare

values about the origin (x=1,y =1) illustrates

the situation:

(44,8),(22,6),(89,23),(37,11)

(14,6),(7,11),(24,16),(12,21);

(87,9),(36,8)

(23,7),(9,12)

Perhaps it may be of interest to examine a detailed example

of how to calculate issquare values at given x,y locations

in pm2, and how to use two sets of these numbers to find two

factors of A:

The value A = 199711 = 41*4871 is found at (27,206) and

(75,31). The first set of coordinates is odd-even and so

calculation of issquare begins at (14,6) on the above chart.

It is not too difficult to figure out what the right values

should be if one just follows the linear progressions of

all even rows and all odd columns to the target.

R1 = 14 + (24 -14)*(27 -1)/2 + (23-14)*(206-2)/2 = 1062,

S1 * S1 = 21*R1*R1 - 5*A,

S1 = 4763;

R2 = 6 + (16 -6)*(27 -1)/2 + (7-6)*(206-2)/2 = 238;

S2 = 437;

The second set of coordinates is odd-odd and so

calculation of issquare begins at (44,8) on the above chart.

R3 = 44 + (89-44)*(75 -1)/2 + (87-44)*(31-1)/2 = 2354,

S3 = 1074 ;

R4 = 8+ (23-8)*(75 -1)/2 + (9-8)*(31-1)/2 = 578

S4 = 2453;

Next,three of the S,R pairs are needed to attempt

the factorization, first calculating:

(R4 +R2)/(S4 +S2) = 816/2890 = 24/85;

also calculating (R4 -R2) = 340 and (S4 -S2) = 1996

and then checking to see if either of these last values

is divisible by either the numerator or denominator

from the first calculation. This divisibility requirement

appears to be a necessary but not sufficient condition

for a solution to exist, and since 340 mod 85 = 0,

we can try GCD(A, 816*R1 +/- 340*(24/85)*S1) to see if

it will yield anything. One factor (41) is found in this

attempt. If none had been found perhaps a different mix

of the S,R pairs would yield something. It is unclear if

a solution always has to exist or not.

In any event the simple reliable factoring method for all

composites of A = 5*x*x + 5*x*y + y*y, is replaced with

a convoluted variant for A = 25*x*x + 25*x*y + y*y that

must include a proportion other than 1:1 to work , and

even with that may still not yield any factors. Here the

proportion is 24/85, but a wide variety is to be expected,

changing for each new composite being examined.

Aldrich Stevens