## Strange blossoms:

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• Speculations concerning binary quadratic forms and issquare functions: The primemine programs that were posted at this PN website needed no reference to
Message 1 of 1 , Apr 6, 2011
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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
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