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• This is a Prime Chain of 161 consecutive terms, including 155 distinct primes, consisting of the output of four well-known prime-producing equations that
Message 1 of 1 , Jul 21, 2008
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This is a Prime Chain of 161 consecutive terms, including
155 distinct primes, consisting of the output of
four well-known prime-producing equations that
alternate sequentially within a procedural expression
of a single polynomial. The equations are:

-(x^5 - 133x^4 + 6729x^3 - 158379x^2 = 1720294x -6823316)/4,
42x^3 + 270x^2 -2643x + 250703,
x^4 - 97x^3 + 3294x^2 -45458x + 213589,
-(3x^3 - 183x^2 + 3318x - 18757)

1705829,250703,213589,
18757,1313781,224579,171329,15619,991127,199247,
135089,12829,729173,174959,104323,10369,519643,151967,78509,8221,
355049,130523,57149,6367,228581,110879,39769,4789,134077,93287,
25919,3469,65993,77999,15173,2389,19373,65267,7129,1531,
-10181,55343,1409,877,-26539,48479,-2341,409,-33073,44927,
-4451,109,-32687,44939,-5227,-41,-27847,48767,-4951,-59,
-20611,56663,-3881,37,-12659,68879,-2251,229,-5323,85667,
-271,499,383,107279,1873,829,3733,133967,4019,1201,
4259,165983,6029,1597,1721,203579,7789,1999,-3923,247007,
9209,2389,-12547,296519,10223,2749,-23887,352367,10789,3061,
-37571,414803,10889,3307,-53149,484079,10529,3469,-70123,560447,
9739,3529,-87977,644159,8573,3469,-106207,735467,7109,3271,
124351,834623,5449,2917,-142019,941879,3719,2389,-158923,1057487,
2069,1669,-174907,1181699,673,739,-189977,1314767,-271,-419,
204331,1456943,-541,-1823,-218389,1608479,109,-3491,-232823,1769627,
1949,-5441,-248587,1940639,5273,-7691,-266947,2121767,10399,-10259,
-289511

The Pascal procedure below should run as an import in several
programming
environments as is, and produce the provided prime chain.

procedure Ndegrees7;
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 := 23;
a[0] := 1705829 { FIRST TERM OF PRIME CHAIN};
writeln('1');
writeln(trunc(a[0]));
writeln;
nh := 1;

a[1] := 250703; ;a[2] := 213589;a[3] := 18757 ;
a[4] := 1313701; a[5] := 224579; a[6] :=171329;
a[7] := 15619; a[8] := 991127; a[9] := 199247;
a[10] := 135089 ;a[11] := 12829 ;
a[12] := 729173 ; a[13] := 174959; a[14] := 104323 ;
a[15] := 10369; a[16] := 519643; a[17] := 151967;
a[18] := 78509;a[19] := 8221; a[20] := 355049 ;
a[21] :=130523; a[22] := 57149 ; a[23] := 6367 ;

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 = 4 THEN A[I] := abs(A[I]); {}
IF NH = 5 THEN A[I] := abs(A[I]); {}
IF NH = 6 THEN A[I] := abs(A[I]); {}
IF NH = 7 THEN A[I] := abs(A[I]); {}
IF NH = 9 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 = 13 THEN A[I] := abs(A[I]); {}
IF NH = 14 THEN A[I] := abs(A[I]); {}
IF NH = 15 THEN A[I] := abs(A[I]); {}
IF NH = 17 THEN A[I] := abs(A[I]); {}
IF NH = 19 THEN A[I] := abs(A[I]); {}
IF NH = 22 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];

if odd(ct ) then a[23] := -a[23];{}
for i := N-1 downto 1 do
begin
if i = 22 then if ct mod 2 = 1 then a[22] := -a[22];{}
if ct > 0 then if i = 19 then if
ct mod 4 = 0 then a[19] := -a[19];{}
if i = 19 then if ct mod 4 = 3 then a[19] := -a[19];{}
if i = 17 then if ct mod 2 = 0 then a[17] := -a[17];{}
if i = 15 then if ct mod 2 = 1 then a[15] := -a[15];{}
if i = 14 then if ct mod 4 = 1 then a[14] := -a[14];{}
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 = 11 then if ct mod 2 = 1 then a[11] := -a[11];{}
if i = 10 then if ct mod 4 <> 1 then a[10] := -a[10];{}
if i = 9 then if ct mod 2 = 1 then a[9] := -a[9];{}
if i = 7 then if ct mod 4 = 1 then a[7] := -a[7];{}
if i = 7 then if ct mod 4 = 0 then a[7] := -a[7];{}
if i = 6 then if ct mod 4 = 1 then a[6] := -a[6];{}
if i = 6 then if ct mod 4 = 0 then a[6] := -a[6];{}
if i = 5 then if ct mod 2 = 0 then a[5] := -a[5];{}
if i = 4 then if ct mod 4 = 0 then a[4] := -a[4];{}
if i = 4 then if ct mod 4 = 3 then a[4] := -a[4];{}
if i = 3 then if ct mod 2 = 1 then a[3] := -a[3];{}

ab2 := a[i] + a[i-1] ;
a[i] := ab1;
ab1 := ab2;
end;

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