novel primality principle pondered
- Actually, I was looking for a decidedly non-Fibonacci way of
ordering an infinite series of valid fourth degree +1 mod 10
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
I have also been unable to find any counterexamples for the pattern
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
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
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
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).