■初等関数研究所■

[0001 ガンマ関数 2019/02/09(土) 17:29:38.68 ID:xWYHJ5Wt]

初等関数（しょとうかんすう、英: Elementary function）とは、
実数または複素数の1変数関数で、代数関数、指数関数、対数関数、
三角関数、逆三角関数および、それらの合成関数を作ることを
有限回繰り返して得られる関数のことである

ガンマ関数、楕円関数、ベッセル関数、誤差関数などは初等関数でない
初等関数のうちで代数関数でないものを初等超越関数という
双曲線関数やその逆関数も初等関数である

初等関数の導関数はつねに初等関数になる

[0662 １３２人目の素数さん 2019/06/14(金) 15:09:28.77 ID:NJp8drOy]

7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0663 １３２人目の素数さん 2019/06/14(金) 15:10:05.72 ID:NJp8drOy]

7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0664 １３２人目の素数さん 2019/06/14(金) 15:10:39.91 ID:NJp8drOy]

7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0665 １３２人目の素数さん 2019/06/14(金) 15:11:18.18 ID:NJp8drOy]

7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0666 １３２人目の素数さん 2019/06/14(金) 15:11:55.23 ID:NJp8drOy]

7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0667 １３２人目の素数さん 2019/06/14(金) 15:12:31.38 ID:NJp8drOy]

7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0668 １３２人目の素数さん 2019/06/14(金) 15:13:05.80 ID:NJp8drOy]

7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0669 １３２人目の素数さん 2019/06/14(金) 15:17:36.98 ID:NJp8drOy]

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Rock54: Caution(BBR-MD5:1341adc37120578f18dba9451e6c8c3b)

[0670 １３２人目の素数さん 2019/06/14(金) 15:21:09.51 ID:NJp8drOy]

7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0671 １３２人目の素数さん 2019/06/14(金) 15:21:49.16 ID:NJp8drOy]

7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0672 １３２人目の素数さん 2019/06/14(金) 16:07:45.51 ID:nclKuWN4]

7 * 8 [1] : 4 , 6 , 4 , 9
7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0673 １３２人目の素数さん 2019/06/14(金) 16:08:36.93 ID:nclKuWN4]

7 * 8 [1] : 4 , 6 , 4 , 9
7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0674 １３２人目の素数さん 2019/06/14(金) 16:09:22.25 ID:nclKuWN4]

7 * 8 [1] : 4 , 6 , 4 , 9
7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0675 １３２人目の素数さん 2019/06/14(金) 16:10:03.31 ID:nclKuWN4]

7 * 8 [1] : 4 , 6 , 4 , 9
7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0676 １３２人目の素数さん 2019/06/14(金) 16:10:39.38 ID:nclKuWN4]

7 * 8 [1] : 4 , 6 , 4 , 9
7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0677 １３２人目の素数さん 2019/06/14(金) 16:12:33.63 ID:nclKuWN4]

7 * 8 [1] : 4 , 6 , 4 , 9
7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0678 １３２人目の素数さん 2019/06/14(金) 16:13:25.15 ID:nclKuWN4]

7 * 8 [1] : 4 , 6 , 4 , 9
7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0679 １３２人目の素数さん 2019/06/14(金) 16:19:20.12 ID:nclKuWN4]

7 * 8 [1] : 4 , 6 , 4 , 9
7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0680 １３２人目の素数さん 2019/06/14(金) 16:20:33.71 ID:nclKuWN4]

7 * 8 [1] : 4 , 6 , 4 , 9
7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0681 １３２人目の素数さん 2019/06/14(金) 16:21:31.79 ID:nclKuWN4]

7 * 8 [1] : 4 , 6 , 4 , 9
7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352

[0682 １３２人目の素数さん 2019/06/14(金) 16:25:18.87 ID:nclKuWN4]

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[0683 １３２人目の素数さん 2019/06/14(金) 16:25:57.56 ID:nclKuWN4]

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[0684 １３２人目の素数さん 2019/06/14(金) 16:26:32.18 ID:nclKuWN4]

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[0685 １３２人目の素数さん 2019/06/14(金) 16:27:21.01 ID:nclKuWN4]

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[0686 １３２人目の素数さん 2019/06/14(金) 16:27:54.02 ID:nclKuWN4]

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[0687 １３２人目の素数さん 2019/06/14(金) 16:30:20.34 ID:nclKuWN4]

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[0688 １３２人目の素数さん 2019/06/14(金) 16:31:00.13 ID:nclKuWN4]

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[0689 １３２人目の素数さん 2019/06/14(金) 16:31:54.90 ID:nclKuWN4]

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[0690 １３２人目の素数さん 2019/06/14(金) 16:32:30.25 ID:nclKuWN4]

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[0691 １３２人目の素数さん 2019/06/14(金) 16:34:03.44 ID:nclKuWN4]

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[0692 １３２人目の素数さん 2019/06/14(金) 16:34:49.69 ID:nclKuWN4]

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[0693 １３２人目の素数さん 2019/06/14(金) 16:35:16.09 ID:nclKuWN4]

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[0694 １３２人目の素数さん 2019/06/14(金) 17:45:26.49 ID:2yYON5Ol]

■ドミノタイリング
https://ja.wikipedia.org/wiki/%E3%83%89%E3%83%9F%E3%83%8E%E3%82%BF%E3%82%A4%E3%83%AA%E3%83%B3%E3%82%B0

[0695 １３２人目の素数さん 2019/06/14(金) 17:59:02.11 ID:2yYON5Ol]

■Die Kasteleyn-Fisher-Temperley-Formel fur die Anzahl der Domino ...
http://oops.uni-oldenburg.de/1773/1/Bachelorarbeit%20von%20Ina%20Lammers.pdf

[0696 １３２人目の素数さん 2019/06/14(金) 18:03:04.31 ID:2yYON5Ol]

■平面充填（へいめんじゅうてん）

平面内を有限種類の平面図形（タイル）で隙間なく敷き詰める操作である
敷き詰めたタイルからなる平面全体を平面充填形という

平面敷き詰め、タイル貼り、タイリング (tiling) 、テセレーション
(tessellation) ともいう
ただし「平面」を明言しない場合は、曲面充填や、
場合によっては2次元以外の空間の充填を含む
広義のテセレーション等については、空間充填を参照
平面充填は広義の空間充填の一種で、2次元ユークリッド空間の
充填である

多面体は多角形による球面充填（曲面充填の一種）と
考えることができる
そのため、多角形による平面充填は多面体と共通点が多く、
便宜上多面体に含めて論じられることもある

[0697 １３２人目の素数さん 2019/06/14(金) 18:05:15.95 ID:2yYON5Ol]

ルジンの問題（Luzin - のもんだい）とは、
正方形に関してニコライ・ルジン (Nikolai Luzin) が考えた問題である

「任意の正方形を、2個以上の全て異なる大きさの正方形に分割できるか」
という問題であり、ルジンはこの問題の解は存在しないと予想したが、
その後幾つかの例が発見された

2, 4, 6, 7, 8, 9, 11, 15, 16, 17, 18, 19, 24, 25, 27, 29, 33, 35, 37, 42, 50 の
計21枚の正方形

[0698 １３２人目の素数さん 2019/06/14(金) 21:32:36.32 ID:qDoAJG0x]

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[0699 １３２人目の素数さん 2019/06/14(金) 21:33:15.37 ID:qDoAJG0x]

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[0700 １３２人目の素数さん 2019/06/14(金) 21:36:48.02 ID:qDoAJG0x]

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[0701 １３２人目の素数さん 2019/06/14(金) 21:37:37.30 ID:qDoAJG0x]

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[0702 １３２人目の素数さん 2019/06/14(金) 21:38:26.70 ID:qDoAJG0x]

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[0703 １３２人目の素数さん 2019/06/14(金) 21:39:11.18 ID:qDoAJG0x]

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[0704 １３２人目の素数さん 2019/06/14(金) 21:41:07.52 ID:qDoAJG0x]

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[0705 １３２人目の素数さん 2019/06/14(金) 21:41:45.87 ID:qDoAJG0x]

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[0706 １３２人目の素数さん 2019/06/14(金) 21:42:18.57 ID:qDoAJG0x]

短小有利、包茎異臭

[0707 １３２人目の素数さん 2019/06/14(金) 21:42:59.30 ID:qDoAJG0x]

短小有利、包茎異臭

[0708 １３２人目の素数さん 2019/06/15(土) 08:40:19.91 ID:7OS3fezI]

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[0709 １３２人目の素数さん 2019/06/15(土) 08:41:02.48 ID:7OS3fezI]

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[0710 １３２人目の素数さん 2019/06/15(土) 08:41:32.32 ID:7OS3fezI]

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[0711 １３２人目の素数さん 2019/06/15(土) 08:42:07.64 ID:7OS3fezI]

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[0712 １３２人目の素数さん 2019/06/15(土) 08:42:52.77 ID:7OS3fezI]

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[0713 １３２人目の素数さん 2019/06/15(土) 09:34:54.78 ID:JxgOS9Yj]

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[0714 １３２人目の素数さん 2019/06/15(土) 09:35:21.27 ID:JxgOS9Yj]

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[0715 １３２人目の素数さん 2019/06/15(土) 09:36:03.58 ID:JxgOS9Yj]

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[0716 １３２人目の素数さん 2019/06/15(土) 09:36:45.47 ID:JxgOS9Yj]

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[0717 １３２人目の素数さん 2019/06/15(土) 09:37:15.23 ID:JxgOS9Yj]

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[0718 １３２人目の素数さん 2019/06/15(土) 09:37:53.89 ID:JxgOS9Yj]

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[0719 １３２人目の素数さん 2019/06/15(土) 09:38:38.68 ID:JxgOS9Yj]

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[0720 １３２人目の素数さん 2019/06/15(土) 09:39:14.82 ID:JxgOS9Yj]

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[0721 １３２人目の素数さん 2019/06/15(土) 09:41:58.44 ID:JxgOS9Yj]

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[0722 １３２人目の素数さん 2019/06/15(土) 09:42:37.00 ID:JxgOS9Yj]

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[0723 １３２人目の素数さん 2019/06/15(土) 16:59:06.66 ID:IIziHdYA]

Π[j=1 to n]Π[k=1 to n]{4cos^2 πj/(2n+1)+4cos^2 πk/(2n+1)}

[0724 １３２人目の素数さん 2019/06/15(土) 18:02:38.57 ID:IKO44kqW]

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[0725 １３２人目の素数さん 2019/06/15(土) 18:03:28.92 ID:IKO44kqW]

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[0726 １３２人目の素数さん 2019/06/15(土) 18:04:01.90 ID:IKO44kqW]

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[0727 １３２人目の素数さん 2019/06/15(土) 18:04:38.69 ID:IKO44kqW]

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[0728 １３２人目の素数さん 2019/06/15(土) 18:05:11.50 ID:IKO44kqW]

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[0729 １３２人目の素数さん 2019/06/15(土) 18:05:46.07 ID:IKO44kqW]

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[0730 １３２人目の素数さん 2019/06/15(土) 18:06:30.42 ID:IKO44kqW]

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[0731 １３２人目の素数さん 2019/06/15(土) 18:07:10.56 ID:IKO44kqW]

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[0732 １３２人目の素数さん 2019/06/15(土) 18:10:28.95 ID:IKO44kqW]

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[0733 １３２人目の素数さん 2019/06/15(土) 18:11:00.17 ID:IKO44kqW]

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[0734 １３２人目の素数さん 2019/06/15(土) 18:54:15.00 ID:u9iyV9ev]

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[0735 １３２人目の素数さん 2019/06/15(土) 18:55:01.03 ID:u9iyV9ev]

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[0736 １３２人目の素数さん 2019/06/15(土) 18:55:39.93 ID:u9iyV9ev]

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[0737 １３２人目の素数さん 2019/06/15(土) 18:56:12.08 ID:u9iyV9ev]

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[0738 １３２人目の素数さん 2019/06/15(土) 18:56:56.78 ID:u9iyV9ev]

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[0739 １３２人目の素数さん 2019/06/15(土) 18:57:29.50 ID:u9iyV9ev]

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[0740 １３２人目の素数さん 2019/06/15(土) 18:58:06.58 ID:u9iyV9ev]

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[0741 １３２人目の素数さん 2019/06/15(土) 18:58:41.36 ID:u9iyV9ev]

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[0742 １３２人目の素数さん 2019/06/15(土) 18:59:18.55 ID:u9iyV9ev]

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[0743 １３２人目の素数さん 2019/06/15(土) 18:59:56.08 ID:u9iyV9ev]

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[0744 １３２人目の素数さん 2019/06/15(土) 19:00:41.92 ID:u9iyV9ev]

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[0745 １３２人目の素数さん 2019/06/15(土) 19:01:18.18 ID:u9iyV9ev]

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[0746 １３２人目の素数さん 2019/06/15(土) 19:43:32.42 ID:IIziHdYA]

縦3マス、横4マスの12マスのうちランダムに選ばれた
2マスにそれぞれ宝が眠っている
AEIBFJ…の順で縦に宝を探していく方法をとるP君と、
ABCDEFGH…の順で横に宝を探していく方法をとるQ君が、
同時に地点Aから探索を開始した
どっちの方が有利？

ABCD
EFGH
I JK L

P1st Q1st even
[1,] 0 0 1
[2,] 4 5 6
[3,] 26 27 13
[4,] 84 83 23
[5,] 203 197 35
[6,] 413 398 50
[7,] 751 722 67
[8,] 1259 1210 87
[9,] 1986 1910 109
[10,] 2986 2875 134

完全追尾型多項式が完成しました

宝の個数は2

P1st={12n^4+28n^3-42n^2-52n-3(-1)^n+51}/48

Q1st={12n^4+20n^3-18n^2-20n-3(-1)^n+3}/48

even={10n^2+8n+(-1)^n-9}/8

■Wolframに入力すると既約分数表示になるので御注意

P1st／Q1st

=8(n-1){(n-2)n-6}/{2n(n+2)(6n^2-2n-5)-3(-1)^n+3}+1

[0747 １３２人目の素数さん 2019/06/15(土) 19:44:17.04 ID:IIziHdYA]

P1stとQ1stは、『宝一つの時の自陣当たり数』の二乗と
それぞれの差分を表す関数の和で求められる

■P1stを求める

宝一つの時の自陣当たり数

n(n+1)/2-1　……�@

P1stは�@^2と差分の和

差分は0 0 1 3 7 13 22 34 50 70 95 125 161 203
252 308 372 444 525 615……

それを表す関数

(4n^3-6n^2-4n-3(-1)^n+3)/48　……�A

計算知能で�@^2+�Aを入力すると

P1st={12n^4+28n^3-42n^2-52n-3(-1)^n+51}/48

■Q1stを求める

宝一つの時の自陣当たり数

n(n+1)/2-1　……�@

Q1stは�@^2と差分の和

差分は0 1 2 2 1 -2 -7 -15 -26 -41 -60 -84 -113
-148 -189……

それを表す関数は　

(-4n^3+18n^2+28n-3(-1)^n-45)/48　……�B

計算知能で�@^2+�Bを入力すると

Q1st={12n^4+20n^3-18n^2-20n-3(-1)^n+3}/48

■evenを求める

evenは、n(n+1)-1と同着数の和

同着数は1 2 4 6 9 12 16 20 25……

これを表す関数は　{2n^2-1+(-1)^(n)}/8　……�C

n(n+1)-1　……�D

計算知能で�C+�Dを入力すると

even={10n^2+8n+(-1)^n-9}/8

[0748 １３２人目の素数さん 2019/06/15(土) 19:44:43.84 ID:IIziHdYA]

P1st Q1st even
[1,] 0 0 1
[2,] 4 5 6
[3,] 26 27 13
[4,] 84 83 23
[5,] 203 197 35
[6,] 413 398 50
[7,] 751 722 67
[8,] 1259 1210 87
[9,] 1986 1910 109
[10,] 2986 2875 134
[11,] 4320 4165 161
[12,] 6054 5845 191
[13,] 8261 7987 223
[14,] 11019 10668 258
[15,] 14413 13972 295
[16,] 18533 17988 335
[17,] 23476 22812 377
[18,] 29344 28545 422
[19,] 36246 35295 469
[20,] 44296 43175 519

Table[(12n^4+28n^3-42n^2-52n-3(-1)^n+51)/48,{n,1,20}]

Table[(12n^4+20n^3-18n^2-20n-3(-1)^n+3)/48,{n,1,20}]

Table[(10n^2+8n+(-1)^n-9)/8,{n,1,20}]

[0749 １３２人目の素数さん 2019/06/15(土) 19:46:33.87 ID:IIziHdYA]

2n x 2n の正方形を
1 x 2 のドミノで埋める場合の数を考えます

たとえば、2x2の正方形を1x2のドミノで埋める場合の数は、2通りです

4x4の正方形を1x2のドミノで埋める場合の数は、36通りです

一般に、n=0,1,2,3,,,,のとき、
1, 2, 36, 6728, 12988816, 258584046368,,,
となり、一般項は、

Π[j=1 to n]Π[k=1 to n]{4cos^2 πj/(2n+1)+4cos^2 πk/(2n+1)}

となるようなのですが、
どのようにその公式が導かれるのでしょうか?

[0750 １３２人目の素数さん 2019/06/15(土) 19:47:24.25 ID:IIziHdYA]

wikipedia
https://en.wikipedia.org/wiki/Domino_tiling
によると
Temperley & Fisher (1961) and Kasteleyn (1961)
によって独立に発見されたとある
多分元論文は
Temperley, H. N. V.; Fisher, Michael E. (1961),
"Dimer problem in statistical mechanics-an exact result",
Philosophical Magazine, 6 (68): 1061-1063, doi:10.1080/14786436108243366

Kasteleyn, P. W. (1961), "The statistics of dimers on a lattice. I.
The number of dimer arrangements on a quadratic lattice", Physica,
27 (12): 1209-1225, Bibcode:1961Phy....27.1209K, doi:10.1016/0031-8914(61)90063-5.

原論文読むのが早い

[0751 １３２人目の素数さん 2019/06/15(土) 19:48:11.89 ID:IIziHdYA]

これに証明載ってるかも
https://inis.iaea.org/collection/NCLCollectionStore/_Public/38/098/38098203.pdf?r=1&;r=1

Section2
A famous result of Kasteleyn [8] and Temperley and Fisher [18] counts the
number of domino tilings of a chessboard (or any other rectangular region).
In this section we explain Kasteleyn's proof.


「ドミノによるタイル張り」(京大･理)　36p.
http://www.ms.u-tokyo.ac.jp/~kazushi/proceedings/domino.pdf

「長方形領域のドミノタイル張りについて」(青学大･理工)　17p.
http://www.gem.aoyama.ac.jp/~kyo/sotsuken/2010/fujino_sotsuron_2010.pdf

[0752 １３２人目の素数さん 2019/06/15(土) 19:48:28.87 ID:IIziHdYA]

Table[{1-n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n-7)(n-8)(n-9)(n-10)(n-11)(n-12)/13!}/4,{n,0,13}]

Table[(1-C(0,n-13))/4,{n,0,13}]

[0753 １３２人目の素数さん 2019/06/15(土) 19:59:39.17 ID:j0UIFbY0]

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[0754 １３２人目の素数さん 2019/06/15(土) 20:00:38.15 ID:j0UIFbY0]

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[0755 １３２人目の素数さん 2019/06/15(土) 20:01:05.63 ID:j0UIFbY0]

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[0756 １３２人目の素数さん 2019/06/15(土) 20:01:56.99 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0757 １３２人目の素数さん 2019/06/15(土) 20:02:36.46 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0758 １３２人目の素数さん 2019/06/15(土) 20:03:17.62 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0759 １３２人目の素数さん 2019/06/15(土) 20:04:01.43 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0760 １３２人目の素数さん 2019/06/15(土) 20:04:41.30 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0761 １３２人目の素数さん 2019/06/15(土) 20:05:28.20 ID:j0UIFbY0]

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sgssl;slg;ld;l

[0762 １３２人目の素数さん 2019/06/15(土) 20:06:05.50 ID:j0UIFbY0]

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sgssl;slg;ld;l