ordering an infinite series of valid fourth degree +1 mod 10

sequences.

Embedded in the third and fourth instances of +/- 1 mod 10

quadratic sequences associated with the primes described below

and indeed for all primes +1 mod 10, there appears to one of

these fourth degree sequences that starts with the quadratic's

term numbers 1,2,5,10,17.

(For 881 in the third instance: 881^2 782611 802021 834571

880561; For 881 in the fourth instance: 881^2 788551 825781

888031 975601) If this is true, it may be the basis for a valid

rule for a different variety of infinite series.

Certain other invariance in the patterns below then suggested an

interesting primality principle, and several related conjectures.

There definitely does seem to be an unexpected connection between

the occurrence or non-occurrence of +/- 1 mod 10 quadratic

sequences that start with a (test#)^2 and :

(1) a conceptually simple (but difficult to implement) way to

determine if the test# is composite or prime.

(2) the presence or absence of factors ending in 3 or 7 in the

test# (also difficult to implement).

Test numbers are limited to integers ending in 1 or 9.

Begin with the square of an integer (T) to be tested for primality.

If the first increment (I) of the fourth +/- mod 10 quadratic

sequence emanating from this square is known then:

If I/10 > T , then T is prime,

Else If I/10 < T, then T is composite

The test#'s listed below are followed by the first increment/10

in their first four instances where valid +/- 1 mod 10 quadratic

sequences are found :

881^2 : 397, 441, 645, 1239

3001^2: 1489, 1501, 2953, 3051

3251^2: 1626, 1639, 3201, 3303

In each instance, the fourth number is higher than the test# (T),

which is prime:

(881 < 1239), (3001 < 3051), (3251 < 3303)

I have been unable to find any counterexamples to this type of

pattern.

I have also been unable to find any counterexamples for the pattern

for composites:

3781: (19*199)^2 : 1692, 1891, 1905, 1923

5611: (31*181)^2: 2533, 2625, 2806, 2823

In each instance, the fourth number is lower than the test # (T) ..

Therefore, if the pattern holds up, establishing that an integer

ending in one is composite is again a matter of checking the

fourth instance of the (first increment)/10. If the fourth

occurrence is < T then the test number is composite:

(3781 > 1923), (5611 > 2823).

The same principle of primality seems to apply to test#'s ending

in 9.

229^2: 103, 115, 159, 345

209: (19*11)^2 : 94, 99, 105, 117

To illustrate (2) above, let us start with the square of the

prime test# 881 , then calculate the first increment from

that to the next member of the sequence that we find gratis

for every test# at 10*(T + 1)/2. (For 881,this is the second

instance a valid +/-mod 10 quadratic)

Thus 10*(881 +1)/2 = 4410. Next add: 881 + 4410 = 5291. This

is the second term of the quadratic sequence. Then add the

constant increment 10 to the first increment to get 4420,

and use this to get 9711, the third term of the quadratic

sequence. Finally we get the sequence 881, 5291, 9711 .. ,

the first three terms of a valid +/-mod 10 quadratic.

If the test# had been 901 = 17*53, then the first increment

of the quadratic, 4510,would not start a valid +/- mod 10

quadratic sequence. There are factors of 17 and 53 in this

sequence.

In examing these patterns, it is evident certain other

conjectures concerning primality and the location of valid

+/- mod 10 quadratic sequences relating to primes and composites

are likely true:

1) There is always a valid sequence (G) beginning with a first

increment of 10*(test# +1)/2 for test numbers with factors

ending exclusively in 1 or 9,and never one for test numbers with

factors ending in 3 or 7.

2) There is floor value (F) at approximately .4473 * T, below

which no (first increment)/10 can occur: For example, for 229

the floor value is 102.

3) If two (sometimes one) valid quadratics can be found between

the floor value and the gratis quadratic at (test# +1)/2, a

test# can be shown to be composite. For 209 (listed above),

showing that 93 (F) < 94 I/10) and 99 (I/10) < 105 (G) would

be sufficient.

As of the moment, none of the preceding conjectures has been

proven, though I believe counterexamples are unlikely, and if

they are proven, they may not be usable. For those who enjoy

contemplating the nature of primality this does not really

matter because the mysterious thing-in-itself gives gratifying

flashes of insight. Practical application ,however, must await a

better means of calculating the location of all the instances

of +/- mod 10 quadratics (if there is one).