Solving polynomial equations and calculating class numbers of imaginary quadratic fields
•
1 like•5,364 views
The document contains mathematical equations and discussions involving prime numbers, polynomials, and Heegner numbers. It explores properties of prime-generating polynomials and uses formulas to calculate values of eπ√d for different values of d, relating these to class numbers of imaginary quadratic fields. Bounds on Heegner numbers d are derived based on results from Heilbronn and Linfoot.
Recommended
More Related Content
What's hot
What's hot (20)
Similar to Solving polynomial equations and calculating class numbers of imaginary quadratic fields
Similar to Solving polynomial equations and calculating class numbers of imaginary quadratic fields (20)
More from Junpei Tsuji
More from Junpei Tsuji (20)
Recently uploaded
Recently uploaded (20)
Solving polynomial equations and calculating class numbers of imaginary quadratic fields
- 4. Q( p -d)
- 6. p -1 13 = ⇣ 2 + 3 p -1 ⌘ ⇣ 2 - 3 p -1 ⌘ 13 = ⇣ 2 + 3 p -1 ⌘ ⇣ 2 - 3 p -1 ⌘
- 7. 1471 = ⇣ 2 + 3 p -163 ⌘ ⇣ 2 - 3 p -163 ⌘ p -163
- 9. 6 = 2 · 3 = ⇣ 1 + p -5 ⌘ ⇣ 1 - p -5 ⌘ Q( p -5)
- 10. Q( p -5), Q( p -6), Q( p -10), Q( p -13), Q( p -14), Q( p -15) Q( p -15), Q( p -17), Q( p -21), Q( p -22), Q( p -23), · · · Q( p -1), Q( p -2), Q( p -3), Q( p -7), Q( p -11), Q( p -19), Q Q( p -7), Q( p -11), Q( p -19), Q( p -43), Q( p -67), Q( p -163)
- 12. d = -1, -2, -3, -7, -11, -19, -43, -67, -163
- 13. X2 - X + 41 4 1
- 14. 12 – 1 + 41 = 22 – 2 + 41 = 32 – 3 + 41 = 42 – 4 + 41 = 52 – 5 + 41 = 62 – 6 + 41 = 72 – 7 + 41 =
- 15. 82 – 8 + 41 = 92 – 9 + 41 = 102 – 10 + 41 = 112 – 11 + 41 = 122 – 12 + 41 = 132 – 13 + 41 = 142 – 14 + 41 =
- 16. 152 – 15 + 41 = 162 – 16 + 41 = 172 – 17 + 41 = 182 – 18 + 41 = 192 – 19 + 41 = 202 – 20 + 41 = 212 – 21 + 41 =
- 17. 222 – 22 + 41 = 232 – 23 + 41 = 242 – 24 + 41 = 252 – 25 + 41 = 262 – 26 + 41 = 272 – 27 + 41 = 282 – 28 + 41 =
- 18. 292 – 29 + 41 = 302 – 30 + 41 = 312 – 31 + 41 = 322 – 32 + 41 = 332 – 33 + 41 = 342 – 34 + 41 = 352 – 35 + 41 =
- 19. 362 – 36 + 41 = 372 – 37 + 41 = 382 – 38 + 41 = 392 – 39 + 41 = 402 – 40 + 41 = 412 – 41 + 41 = = 412
- 20. () X = 1 q – 1 q = 1 + d 4
- 21. X2 – X + X = 1 X2 – X + X = 1, 2 X2 – X + X = 1, 2, 3, 4, 5 X2 – X + X = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 X2 – X + X = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 X2 – X + X = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …, 40
- 23. Google
- 24. e⇡ p d e⇡ p d = ( )3 + 744 – ( )
- 25. e⇡ p 19 = 963 + 744 - 0.22 . . . e⇡ p 43 = 9603 + 744 - 0.00022 . . . e⇡ p 67 = 52803 + 744 - 0.0000013 . . .
- 26. e⇡ p 163 163 = 6403203 + 744 - 0.00000000000075 . . . 12 digits
- 27. e⇡ p d () X = 1 q – 1
- 30. Q( p -d) () (=)) ((=) (1912 ) (1913 ) X = 0, · · · , q - 2 X2 + X + q tsujimotter http://tsujimotter.hatenablog.com/entry/prime-generating-polynomials
- 31. j(⌧) = 1 q + 744 + 196884q + 21493760q3 + · · · j- q- ⌧ = 1 + p -d 2 q = - 1 e⇡ p d e⇡ p d = -j ✓ 1 + p -d 2 ◆ + 744 - 196884 ✓ 1 e⇡ p d ◆ - 21493760 ✓ 1 e⇡ p d ◆3 - · · ·
- 32. Romantic formula 1 ⇡ = 12 6403203/2 1X k=0 (6k)!(163 · 3344418k + 13591409) (3k)!(k!)3(-640320)3k j ✓ 1 + p -163 2 ◆ = -6403203 [Chudnovsky brothers 1989]
- 33. Heegner Number Q( p -d) Heegner Number h log " - 32 21 ⇡ p d < e ⇡ p d 100 h0 log "0 - 80 33 ⇡ p d < e ⇡ p d 100 ---- (1) ---- (2) (1), (2) |b log " + b0 log "0 | < e- B B < ⇣ 4n2 -1 l2n log A ⌘(2n+1)2 < 10250 ⇣ B = 140 p d ⌘ ) d < 10500 (A)
- 34. Heegner Number Heilbronn, Linfoot 10 Heegner Number d > e1000000 (B) d (B)(A) d > e1000000) d < 10500 d (QED) tsujimotter http://tsujimotter.hatenablog.com/entry/class-numbers-of-imaginary-quadratic-fields