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• An open problem is to find out polynomials of degree 2 giving L distinct primes when p 41 and 40
Message 1 of 1 , Mar 26, 2006
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An open problem is to find out polynomials of degree 2 giving L
distinct primes when p > 41 and 40 < L < p-1.

The following consderations should reduce the amount of search time.

Algebraic solution to prime generating quadratics

X^2 + x + P will be a prime rich polynomial if

x^2 + x + P is prime mod q for
q = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,53, 59, 61,
67, 71, 73, 79, 83, 89, and 97.

In addition, if k = x^2 + x + P satisfies the above requirement,
then any factors of k must be greater than 97.

Thus, we can generate the best candidates for the prime generating
solving for which numbers P make x^2 + x + P a prime polynomial mod q for 1
< q < 100.

Solve as follows:

x^2 + x + p = 0 mod q

x = (- 1 + sqrt(1 - 4p) )/2

x^2 + x + p = 0 mod q has no solution

if (1-4 p) is a non square residue.

That is,

x^2 + x + p = 0 mod q if

p = (1 - n)/4 where n is a non square residue.

This leads to the following theorem:

You should recalculate my numbers rather than depend on my
getting them right.

X^2 + x + P is prime rich

if in mod 3,

P = 2

and

in mod 5

P = 1 or 2

and

in mod 7

P 3 or 6 or 4

and

in mod 11

P = 8 or 7 or 4 or 1 or 6

and in mod 13

P = 3 or 12 or 2 or 5 or 8 or 4

and

in mod 17

P = 8 or 16 or 3 or 7 or 2 or 6 or 10 or 1

and

in mod 19

P = 14 or 9 or 3 or 12 or 2 or 16 or 11 or 6 or 10

and

in mod 23

P = 22 or 10 or 15 or 9 or 14 or 8 or 19 or 7 or 1
or 18 or 12

and in mod 29

P = 7 or 14 or 20 or 5 or 12 or 19 or 4 or 11 or 25
or 3 or 10 or 24 or 1 or 8

and in mod 31

P = 15 or 22 or 13 or 5 or 28 or 12 or 27 or 26 or 18
or 10 or 2 or 17 or 9 or 24 or 16

and

in mod 37

P = 9 or 36 or 8 or 26 or 34 or 6 or 15 or 33 or 5
or 14 or 23 or 4 or 13 or 22 or 30 or 11 or 20
or 10

and

in mod 41

P = 20 or 9 or 19 or 18 or 28 or 38 or 7 or 17 or 37
or 16 or 5 or 25 or 4 or 14 or 24 or 34 or 3 or 2
or 12 or 1

and

in mod 43

P = 32 or 21 or 42 or 20 or 9 or 8 or 28 or 17 or 6 or 27
or 26 or 15 or 4 or 36 or 25 or 3 or 35 or 24 or 34
or 12 or 22

and

in mod 47

P = 46 or 33 or 21 or 44 or 20 or 19 or 7 or 30 or 18
or 29 or 40 or 28 or 16 or 39 or 15 or 26 or 14 or 2
or 37 or 13 or 1 or 36 or 24

and

in mod 53

P = 13 or 26 or 52 or 38 or 37 or 10 or 9 or 22 or 35
or 48 or 8 or 21 or 7 or 20 or 6 or 19 or 32 or 45
or 5 or 18 or 17 or 43 or 42 or 28 or 1 or 14

and

in mod 59

P = 44 or 43 or 13 or 42 or 27 or 56 or 41 or 40
or 24 or 9 or 37 or 22 or 7 or 51 or 36 or 50
or 35 or 20 or 5 or 34 or 19 or 4 or 18
or 32 or 2 or 31 or 16 or 1 or 30

and

in mod 61

P = 15 or 14 or 29 or 44 or 13 or 28 or 57 or 11 or 56
or 25 or 40 or 9 or 39 or 54 or 8 or 23 or 38 or 53
or 22 or 52 or 6 or 36 or 20 or 35 or 3 or 18 or 48
or 2 or 17 or 16

and

in mod 67

P = 50 or 33 or 66 or 32 or 15 or 31 or 14 or 64
or 46 or 12 or 27 or 10 or 43 or 26 or 9 or 42
or 41 or 57 or 40 or 23 or 6 or 56 or 39 or 5
or 38 or 21 or 4 or 54 or 53 or 36 or 52 or 18
or 34

and

in mod 71

P = 34 or 33 or 68 or 50 or 67 or 66 or 48 or 30
or 47 or 11 or 28 or 63 or 45 or 27 or 26 or 61
or 43 or 7 or 42 or 24 or 23 or 5 or 58 or 22
or 4 or 21 or 56 or 38 or 20 or 55 or 37 or 19
or 1 or 54 or 36

and

in mod 73

P = 72 or 35 or 16 or 34 or 70 or 15 or 33 or 69
or 50 or 68 or 13 or 12 or 48 or 66 or 11
or 29 or 65 or 10 or 27 or 45 or 8 or 26
or 44 or 62 or 25 or 24 or 42 or 60 or 41
or 4 or 22 or 40 or 3 or 21 or 2 or 38

and

in mod 79

P = 39 or 58 or 38 or 17 or 56 or 36 or 75 or 14
or 33 or 13 or 72 or 52 or 71 or 51 or 31 or 70
or 30 or 69 or 29 or 28 or 8 or 66 or 46 or 6
or 65 or 45 or 25 or 5 or 64 or 24 or 43 or 3
or 62 or 42 or 22 or 41 or 21 or 60 or 40

and

in mod 83

P = 62 or 82 or 61 or 19 or 80 or 59 or 38 or 58
or 37 or 16 or 57 or 15 or 13 or 54 or 33
or 32 or 52 or 31 or 72 or 51 or 30 or 50
or 8 or 70 or 49 or 28 or 7 or 69 or 48
or 6 or 47 or 46 or 25 or 24 or 3 or 65
or 44 or 2 or 22 or 1 or 42

and

in mod 89

P = 44 or 21 or 43 or 64 or 86 or 19 or 41 or 40
or 39 or 61 or 16 or 38 or 60 or 82 or 15 or 37
or 81 or 36 or 80 or 13 or 79 or 34 or 11 or 55
or 32 or 54 or 9 or 53 or 8 or 30 or 52 or 74
or 7 or 29 or 73 or 6 or 5 or 4 or 26 or 48
or 70 or 2 or 24 or 1

and

in mod 97

P = 96 or 47 or 22 or 94 or 21 or 45 or 93 or 44
or 68 or 92 or 43 or 18 or 66 or 90 or 17 or 16
or 88 or 15 or 39 or 63 or 87 or 14 or 86 or 13
or 36 or 60 or 35 or 59 or 83 or 10 or 34 or 58
or 33 or 32 or 56 or 80 or 31 or 6 or 54 or 78
or 5 or 53 or 4 or 28 or 52 or 27 or 2 or 50
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