Beyond Binary Claims: Mathematical Discovery…

[Ken Clements]

If TTP is the total number of twin prime pairs less than or equal to n, then:

| n | TTP | TPπ2 | Ratio (TPπ2/TTP) |

|-----------------|--------------|-------|-------------------|

| 1,050 | 37 | 6 | 0.1622 |

| 10,500 | 215 | 9 | 0.04186 |

| 100,800 | 1,229 | 9 | 0.007323 |

| 1,008,420 | 8,237 | 13 | 0.001578 |

| 10,106,250 | 59,532 | 13 | 0.0002184 |

| 100,018,800 | 440,389 | 15 | 3.406 × 10⁻⁵ |

| 1,039,218,180 | 3,544,751 | 16 | 4.514 × 10⁻⁶ |

| 10,034,584,560 | 27,498,915 | 17 | 6.182 × 10⁻⁷ |

| 100,714,764,150 | 225,845,904 | 18 | 7.970 × 10⁻⁸ |

[Ken Clements]

Here is the progression of TPπn distributions going up the number line:

n = 1050

TPπ2 = 6

TPπ3 = 8

TPπ4 = 2

n = 10,500

TPπ2 = 9

TPπ3 = 14

TPπ4 = 9

TPπ5 = 2

n = 100,800

TPπ2 = 9

TPπ3 = 18

TPπ4 = 17

TPπ5 = 7

n = 1,008,420

TPπ2 = 13

TPπ3 = 23

TPπ4 = 33

TPπ5 = 17

TPπ6 = 9

n = 10,106,250

TPπ2 = 13

TPπ3 = 26

TPπ4 = 54

TPπ5 = 44

TPπ6 = 20

TPπ7 = 2

n = 100,018,800

TPπ2 = 15

TPπ3 = 26

TPπ4 = 93

TPπ5 = 71

TPπ6 = 40

TPπ7 = 9

n = 1,039,218,180

TPπ2 = 16

TPπ3 = 50

TPπ4 = 129

TPπ5 = 113

TPπ6 = 104

TPπ7 = 37

TPπ8 = 2

TPπ9 = 1

n = 10,034,584,560

TPπ2 = 17

TPπ3 = 63

TPπ4 = 172

TPπ5 = 185

TPπ6 = 179

TPπ7 = 95

TPπ8 = 20

TPπ9 = 9

n = 100,714,764,150

TPπ2 = 18

TPπ3 = 81

TPπ4 = 213

TPπ5 = 288

TPπ6 = 294

TPπ7 = 219

TPπ8 = 74

TPπ9 = 9