Prime Chain of 128

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• This is a Prime Chain of 128 terms, including 104 distinct primes, consisting of the output of eight equations that alternate sequentially within a procedural
Message 1 of 1 , Jul 12, 2008
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This is a Prime Chain of 128 terms, including
104 distinct primes, consisting of the output of
eight equations that alternate sequentially within
a procedural expression of a single polynomial.
The equations are either subsequences of
x^2 - 79x + 1601 or transforms
with one exception: 100x^2 -2260x + 12959

The other four distinct equations are
Euhler-derived.:
25x^2 -1185x + 14083,
25x^2 -775x + 6047,
100x^2 -2280x + 13159,
100x^2 -4160x +43427

14083, 14087, 6047, 13159, 12923, 43427,
5297, 12959, 11813, 11827, 4597,
10979, 10753, 39367, 3947, 10799,
9743, 9767, 3347, 8999 ,8783,
35507, 2797, 8839, 7873, 7907,
2297, 7219, 7013, 31847, 1847,
7079, 6203, 6247, 1447, 5639
5443, 28387, 1097, 5519, 4733
4787, 797, 4259 4073, 25127,
547, 4159, 3463, 3527, 347,
3079, 2903, 22067, 197, 2999,
2393, 2467, 97, 2099, 1933,
19207, 47, 2039, 1523, 1607,
47, 1319, 1163, 16547, 97,
1279, 853, 947, 197, 739,
593, 14087, 347, 719, 383,
487, 547, 359, 223, 11827,
797, 359, 113, 227, 1097,
179, 53, 9767, 1447, 199,
43, 167, 1847, 199, 83,
7907, 2297, 239, 173, 307,
2797, 419, 313, 6247, 3347,
479, 503, 647, 3947, 839,
743, 4787, 4597, 919, 1033,
1187, 5297, 1459, 1373, 3527,
6047, 1559

Perhaps its obvious, but I think that I should mention that the
basis for this prime chain in my little procedure at bottom is
the difference equation stack concept. This allows one to
generate all of the values of a polynomial without knowing the
equation providing the first few outputs are known. A simple example
can be illustrated. For instance, if one has the outputs 41, 43, 47
then a difference equation pyramid can be developed that will
generate the same sequence as x^2 - x + 41:
2
2 4
41 43 47
Using my procedure, I would begin with the stack on the
left-hand side of the pyramid and start a "repeat loop".
2 2 2 2
2 + 2 = 4 + 2 = 6 + 2 = 8
41 + 2 = 43 + 4 = 47 + 6 = 53 etc.
A more complex difference equation pyramid is necessary to
generate the rest of the sequence for 11, 61, 281, 911,2311...
(5x^4 - 10x^3 + 20x^2 -15x + 11). The procedure is easily
adjustable to do this for any degree polynomial.
120
240 360
170 410 770
50 220 630 1400
11 61 281 911 2311
Sarting with the left-hand stack as before, the calculations proceed:
120 120 120 120
240 + 120 = 360 + 120 = 480 + 120 = 600
170 + 240 = 410 + 360 = 770 + 480 = 1250
50 + 170 = 220 + 410 = 630 + 770 = 1400
11 + 50 = 61 + 220 = 281 + 630 = 911 etc.

I believe that concept of the difference equation stack probably
predates polynomial nomenclature, though my number theory books
fail to mention the idea at all. The melding of several polynomials
at once with a difference equation stack is not mentioned either,
but it may be easily observed that success in that endeavor will
surely depend heavily on the initial pieces of equations and
transforms chosen, and their ordering, and that certainly,without
modification, the difference equation pyramid will always go to
infinity in that situation and be useless.

I have found that the use of the sign change of the absolute value
function on selected rows can often be salutory in that regard,
and resolve the pyramid to a constant at its apex.
At each row from bottom to top one must experiment with changing
all negative terms to positive with an eye to causing the next row
to be smaller in size, less erratic in pattern, and for the last
digit of all terms to be identical. There may be several ways to get
the pyramid to resolve to a constant,or none at all, and it is
quite possible that there are smaller initial left-hand stacks for
our present polynomial (abs denotes rows to be modified by the
absolute value function)
than this one:
abs 160
0
abs 1820
0
abs 1760
abs 0
abs 59160
abs 0
-1090
abs 125806
62518
abs 29414
16284
-696
abs 23196
-804
4
14083

The Pascal procedure below should run as an import in several
programming
environments as is.

procedure Ndegrees5;
var a : array[0..32] of extended;
ct: longint;
n,nh ,i,j : integer;
ab1,ab2 : extended;
begin
for i := 0 to 32 do
a[i] := 0;
N := 17;
a[0] := 14083{ FIRST TERM OF PRIME CHAIN};
writeln('1');
writeln(trunc(a[0]));
writeln;
nh := 1;
a[1] := 14087 ;a[2] := 6047;
a[3] := 13159 ; a[4] :=12923 ; a[5] := 43427; a[6] := 5297 ;
a[7] := 12959 ; a[8] :=11813 ; a[9] :=11827 ;a[10] := 4597 ;
a[11] :=10979 ; a[12] := 10753 ; a[13] := 39367 ;a[14] := 3947;
a[15] :=10799 ;a[16] := 9743 ;a[17] := 9767 ;
repeat
for i := N downto nh do
begin
a[i] := a[i] - a[i-1] ;
IF NH = 3 THEN A[I] := abs(A[I]); {}
iF NH = 6 THEN A[I] := abs(A[I]); {}
iF NH = 8 THEN A[I] := abs(A[I]); {}
iF NH = 10 THEN A[I] := abs(A[I]); {}
iF NH = 11 THEN A[I] := abs(A[I]); {}
iF NH = 12 THEN A[I] := abs(A[I]); {}
iF NH = 13 THEN A[I] := abs(A[I]); {}
iF NH = 15 THEN A[I] := abs(A[I]); {}
iF NH = 17 THEN A[I] := abs(A[I]); {}
end;
nh := nh + 1;
until nh = n + 2;
ct := 0;
repeat
ct := ct + 1;
ab1 := a[n] + a[n-1];
for i := N-1 downto 1 do
begin
if i = 15 then if ct mod 2 = 0 then a[15] := -a[15];{}
if i = 14 then if ct mod 4 = 0 then a[14] := -a[14];{}
if i = 13 then if ct mod 2 = 0 then a[13] := -a[13];{}
if i = 12 then if ct mod 4 = 0 then a[12] := -a[12];{}
if i = 11 then if ct mod 2 = 0 then a[11] := -a[11];{}
if i = 10 then if ct mod 8 = 2 then a[10] := -a[10];{}
if i = 10 then if ct mod 8 = 0 then a[10] := -a[10];{}
if i = 8 then if ct mod 2 = 1 then a[8] := -a[8];{}
if i = 6 then if ct mod 4 = 2 then a[6] := -a[6];{}
if i = 6 then if ct mod 4 = 3 then a[6] := -a[6];{}
if i = 3 then if ct mod 2 = 0 then a[3] := -a[3];{}

ab2 := a[i] + a[i-1] ;
a[i] := ab1;
ab1 := ab2;
end;
if odd(ct) then a[17] := -a[17];

a[0] := ab1;
writeln(ct + 1);
writeln(trunc(a[0]));{}