A beautifull day
i have written a quadratic sieve algorithm which produces all prime
numbers up to a limit. I find the complete list up to liste_max and some bigger
primes.I examine the three polynoms 4x^2+1, 2x^2+1
and 2x^2-1. All prime numbers can be found on these tree polynoms.
Let p be a prime, then you can find the prime on the following polynom.
2x^2 + 1 p = 1,3 mod 8,
4x^2 + 1 p = 1 mod 4 and
2x^2 - 1 p = 1,7 mod 8.
I hope that my algorithm is faster then the sieve of Eratosthenes,
because the density of the primes is nearly constant concerning the
primes you can found on the polynoms.
I am looking forward for your answer
Best greetings from the primes
Bernhard Helmes
Sourcecode in MuPad:
Find all primes below liste_max
liste_max:=100;
z:={};
siebung:=proc (stelle)
begin
while (stelle<=liste_max) do
erg:=liste[stelle];
erg:=erg /p;
while (erg mod p = 0) do
erg:=erg /p;
end_while;
liste[stelle]:=erg;
stelle:=stelle+p;
end_while;
end_proc;
// 1. Polynom 4*x^2+1
for x from 1 to liste_max do
liste [x]:=4*x^2+1;
end_for;
for x from 1 to liste_max do
p:=liste[x];
if (p>1) then
// print ("x = ", x, "Prim = ", p) ;
z:=z union {p};
siebung (x+p);
siebung (p-x);
end_if;
end_for;
// 2. Polynom 2*x^2+1
for x from 1 to liste_max do
liste [x]:=2*x^2+1;
end_for;
for x from 1 to liste_max do
p:=liste[x];
if (p>1) then
z:=z union {p};
// print ("x = ", x, "Prim = ", p) ;
siebung (x+p);
siebung (p-x);
end_if;
end_for;
// 3. Polynom 2*x^2-1
for x from 1 to liste_max do
liste [x]:=2*x^2-1;
end_for;
for x from 1 to liste_max do
p:=liste[x];
if (p>1) then
z:=z union {p};
// print ("x = ", x, "Prim = ", p) ;
siebung (x+p);
siebung (p-x);
end_if;
end_for;
print (z)
{3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137,
139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199,
211, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283,
293, 307, 311, 313, 317, 337, 349, 353, 373, 379, 401, 419, 431, 449, 457, 461,
467, 491, 523, 541, 547, 557, 569, 577, 593, 599, 601, 607, 617, 631,
641,647,661, 673, 677, 683, 709, 743, 769, 827, 881, 883, 911, 937, 953,
967,1063, 1129, 1151, 1153, 1193, 1217, 1249, 1283, 1291, 1297, 1321,
1459, 1481, 1531, 1549, 1567, 1601, 1607, 1621, 1667, 1693, 1721, 1783,
1801, 1873, 2017, 2069, 2081, 2113, 2179, 2243, 2311, 2333, 2591, 2593,
2633, 2731, 2777, 2857, 2887, 2917, 3041, 3083, 3137, 3361, 3527, 3529,
3697, 3851, 4049, 4051, 4231, 4357, 4483, 4621, 4801, 4817, 4931, 4993,
4999, 5281, 5407, 5477, 5521, 6271, 6337, 6961, 7057, 7069, 7687, 7841,
7937, 8101, 8191, 8713, 8837, 9521, 10369, 10657, 11251, 11551, 12101,
12799, 13121, 13457, 14401, 14449, 15137, 15139, 15377, 15877, 16561,
16901, 16927, 17299, 17957, 18049, 18433, 19207, 19603, 21317, 22501,
24337, 25601, 28901, 30977, 32401, 33857}
--
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