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W2038846491 abstract "I present a new algorithm for computing binomial coefficients modulo <svg style=vertical-align:-0.0pt;width:19.3125px; id=M1 height=16.012501 version=1.1 viewBox=0 0 19.3125 16.012501 width=19.3125 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,15.95)><path id=x32 d=M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z /></g> <g transform=matrix(.012,-0,0,-.012,8.225,7.8)><path id=x1D441 d=M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z /></g> </svg>. The proposed method has an <svg style=vertical-align:-3.27599pt;width:219.9375px; id=M2 height=19.924999 version=1.1 viewBox=0 0 219.9375 19.924999 width=219.9375 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,15.775)><path id=x1D442 d=M745 361q0 -166 -116 -272t-291 -106q-136 0 -225.5 83t-89.5 218q0 161 115.5 272t288.5 111q137 0 227.5 -83t90.5 -223zM643 359q0 126 -56 199.5t-167 73.5q-130 0 -213.5 -104t-83.5 -246q0 -117 58 -190.5t165 -73.5q91 0 160.5 52t103 128.5t33.5 160.5z /></g><g transform=matrix(.017,-0,0,-.017,13.118,15.775)><path id=x28 d=M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z /></g><g transform=matrix(.017,-0,0,-.017,18.999,15.775)><use xlink:href=#x1D441/></g> <g transform=matrix(.012,-0,0,-.012,33.85,7.613)><path id=x33 d=M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
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t60.5 43z /></g><g transform=matrix(.017,-0,0,-.017,113.912,15.775)><path id=x61 d=M433 39l-85 -51q-27 0 -47 20q-16 16 -24 44q-5 -3 -24 -16.5t-27 -18.5t-23 -13.5t-25 -12t-19 -3.5q-52 0 -86 36.5t-34 85.5q0 71 81 99q128 43 155 65v17q0 54 -23 83.5t-62 29.5q-28 0 -45 -19t-29 -66q-7 -23 -29 -23q-15 0 -29 13t-14 30t31 40q25 19 68.5 40.5
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 /></g><g transform=matrix(.017,-0,0,-.017,148.658,15.775)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,154.539,15.775)><use xlink:href=#x1D441/></g><g transform=matrix(.017,-0,0,-.017,169.396,15.775)><path id=x29 d=M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z /></g><g transform=matrix(.017,-0,0,-.017,179.035,15.775)><path id=x2B d=M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z /></g><g transform=matrix(.017,-0,0,-.017,192.787,15.775)><use xlink:href=#x1D441/></g> <g transform=matrix(.012,-0,0,-.012,207.662,7.613)><path id=x34 d=M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z /></g> <g transform=matrix(.017,-0,0,-.017,214,15.775)><use xlink:href=#x29/></g> </svg> preprocessing time, after which a binomial coefficient <svg style=vertical-align:-2.3205pt;width:54.450001px; id=M3 height=15.0875 version=1.1 viewBox=0 0 54.450001 15.0875 width=54.450001 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.138)><path id=x1D436 d=M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5
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q0 -41 38 -68.5t94 -27.5q69 0 106.5 31t37.5 80q0 52 -45 71q-33 13 -98 13z /></g><g transform=matrix(.017,-0,0,-.017,72.537,15.775)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,78.419,15.775)><use xlink:href=#x1D441/></g><g transform=matrix(.017,-0,0,-.017,93.276,15.775)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,102.931,15.775)><use xlink:href=#x22C5/></g><g transform=matrix(.017,-0,0,-.017,110.581,15.775)><use xlink:href=#x4D/></g><g transform=matrix(.017,-0,0,-.017,125.183,15.775)><use xlink:href=#x75/></g><g transform=matrix(.017,-0,0,-.017,134.277,15.775)><use xlink:href=#x6C/></g><g transform=matrix(.017,-0,0,-.017,138.391,15.775)><use xlink:href=#x74/></g><g transform=matrix(.017,-0,0,-.017,143.643,15.775)><use xlink:href=#x69/></g><g transform=matrix(.017,-0,0,-.017,148.012,15.775)><use xlink:href=#x70/></g><g transform=matrix(.017,-0,0,-.017,156.8,15.775)><use xlink:href=#x6C/></g><g transform=matrix(.017,-0,0,-.017,161.101,15.775)><use xlink:href=#x69/></g><g transform=matrix(.017,-0,0,-.017,165.657,15.775)><use xlink:href=#x63/></g><g transform=matrix(.017,-0,0,-.017,172.881,15.775)><use xlink:href=#x61/></g><g transform=matrix(.017,-0,0,-.017,180.037,15.775)><use xlink:href=#x74/></g><g transform=matrix(.017,-0,0,-.017,185.29,15.775)><use xlink:href=#x69/></g><g transform=matrix(.017,-0,0,-.017,189.846,15.775)><use xlink:href=#x6F/></g><g transform=matrix(.017,-0,0,-.017,198.328,15.775)><use xlink:href=#x6E/></g><g transform=matrix(.017,-0,0,-.017,207.626,15.775)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,213.508,15.775)><use xlink:href=#x1D441/></g><g transform=matrix(.017,-0,0,-.017,228.365,15.775)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,234.246,15.775)><use xlink:href=#x29/></g> </svg> time. <svg style=vertical-align:-3.27599pt;width:123.775px; id=M7 height=16.3125 version=1.1 viewBox=0 0 123.775 16.3125 width=123.775 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.162)><use xlink:href=#x4D/></g><g transform=matrix(.017,-0,0,-.017,14.647,12.162)><use xlink:href=#x75/></g><g transform=matrix(.017,-0,0,-.017,23.742,12.162)><use xlink:href=#x6C/></g><g transform=matrix(.017,-0,0,-.017,27.855,12.162)><use xlink:href=#x74/></g><g transform=matrix(.017,-0,0,-.017,33.108,12.162)><use xlink:href=#x69/></g><g transform=matrix(.017,-0,0,-.017,37.477,12.162)><use xlink:href=#x70/></g><g transform=matrix(.017,-0,0,-.017,46.265,12.162)><use xlink:href=#x6C/></g><g transform=matrix(.017,-0,0,-.017,50.566,12.162)><use xlink:href=#x69/></g><g transform=matrix(.017,-0,0,-.017,55.121,12.162)><use xlink:href=#x63/></g><g transform=matrix(.017,-0,0,-.017,62.346,12.162)><use xlink:href=#x61/></g><g transform=matrix(.017,-0,0,-.017,69.502,12.162)><use xlink:href=#x74/></g><g transform=matrix(.017,-0,0,-.017,74.755,12.162)><use xlink:href=#x69/></g><g transform=matrix(.017,-0,0,-.017,79.311,12.162)><use xlink:href=#x6F/></g><g transform=matrix(.017,-0,0,-.017,87.793,12.162)><use xlink:href=#x6E/></g><g transform=matrix(.017,-0,0,-.017,97.091,12.162)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,102.973,12.162)><use xlink:href=#x1D441/></g><g transform=matrix(.017,-0,0,-.017,117.83,12.162)><use xlink:href=#x29/></g> </svg> denotes the time complexity of multiplying two <svg style=vertical-align:-0.1092pt;width:14.9875px; id=M8 height=11.3125 version=1.1 viewBox=0 0 14.9875 11.3125 width=14.9875 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,11.113)><use xlink:href=#x1D441/></g> </svg>-bit numbers, which can range from <svg style=vertical-align:-2.3205pt;width:46.137501px; id=M9 height=18.725 version=1.1 viewBox=0 0 46.137501 18.725 width=46.137501 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,15.775)><use xlink:href=#x1D442/></g><g transform=matrix(.017,-0,0,-.017,13.118,15.775)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,18.999,15.775)><use xlink:href=#x1D441/></g> <g transform=matrix(.012,-0,0,-.012,33.85,7.613)><use xlink:href=#x32/></g> <g transform=matrix(.017,-0,0,-.017,40.188,15.775)><use xlink:href=#x29/></g> </svg> to <svg style=vertical-align:-3.50803pt;width:190.4375px; id=M10 height=16.6 version=1.1 viewBox=0 0 190.4375 16.6 width=190.4375 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.162)><use xlink:href=#x1D442/></g><g transform=matrix(.017,-0,0,-.017,13.118,12.162)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,18.999,12.162)><use xlink:href=#x1D441/></g><g transform=matrix(.017,-0,0,-.017,37.63,12.162)><use xlink:href=#x22C5/></g><g transform=matrix(.017,-0,0,-.017,45.279,12.162)><use xlink:href=#x6C/></g><g transform=matrix(.017,-0,0,-.017,49.58,12.162)><use xlink:href=#x6F/></g><g transform=matrix(.017,-0,0,-.017,58.249,12.162)><use xlink:href=#x67/></g><g transform=matrix(.017,-0,0,-.017,66.205,12.162)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,72.086,12.162)><use xlink:href=#x1D441/></g><g transform=matrix(.017,-0,0,-.017,86.943,12.162)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,96.598,12.162)><use xlink:href=#x22C5/></g><g transform=matrix(.017,-0,0,-.017,104.248,12.162)><use xlink:href=#x6C/></g><g transform=matrix(.017,-0,0,-.017,108.549,12.162)><use xlink:href=#x6F/></g><g transform=matrix(.017,-0,0,-.017,117.218,12.162)><use xlink:href=#x67/></g><g transform=matrix(.017,-0,0,-.017,125.173,12.162)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,131.055,12.162)><use xlink:href=#x6C/></g><g transform=matrix(.017,-0,0,-.017,135.356,12.162)><use xlink:href=#x6F/></g><g transform=matrix(.017,-0,0,-.017,144.025,12.162)><use xlink:href=#x67/></g><g transform=matrix(.017,-0,0,-.017,151.98,12.162)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,157.862,12.162)><use xlink:href=#x1D441/></g><g transform=matrix(.017,-0,0,-.017,172.719,12.162)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,178.6,12.162)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,184.482,12.162)><use xlink:href=#x29/></g> </svg> or better. Thus, the overall time complexity for evaluating <svg style=vertical-align:-0.0pt;width:17.475px; id=M11 height=11.175 version=1.1 viewBox=0 0 17.475 11.175 width=17.475 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,11.113)><path id=x1D440 d=M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z /></g> </svg> binomial coefficients <svg style=vertical-align:-2.3205pt;width:54.450001px; id=M12 height=15.0875 version=1.1 viewBox=0 0 54.450001 15.0875 width=54.450001 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.138)><use xlink:href=#x1D436/></g><g transform=matrix(.017,-0,0,-.017,12.047,12.138)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,17.928,12.138)><use xlink:href=#x1D443/></g><g transform=matrix(.017,-0,0,-.017,28.739,12.138)><use xlink:href=#x2C/></g><g transform=matrix(.017,-0,0,-.017,35.437,12.138)><use xlink:href=#x1D444/></g><g transform=matrix(.017,-0,0,-.017,48.492,12.138)><use xlink:href=#x29/></g> </svg> modulo <svg style=vertical-align:-0.0pt;width:19.3125px; id=M13 height=16.012501 version=1.1 viewBox=0 0 19.3125 16.012501 width=19.3125 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,15.95)><use xlink:href=#x32/></g> <g transform=matrix(.012,-0,0,-.012,8.225,7.8)><use xlink:href=#x1D441/></g> </svg> with <svg style=vertical-align:-2.2932pt;width:135.28751px; id=M14 height=18.875 version=1.1 viewBox=0 0 135.28751 18.875 width=135.28751 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,15.95)><use xlink:href=#x30/></g><g transform=matrix(.017,-0,0,-.017,12.931,15.95)><use xlink:href=#x2264/></g><g transform=matrix(.017,-0,0,-.017,27.634,15.95)><use xlink:href=#x1D444/></g><g transform=matrix(.017,-0,0,-.017,45.415,15.95)><use xlink:href=#x2264/></g><g transform=matrix(.017,-0,0,-.017,60.119,15.95)><use xlink:href=#x1D443/></g><g transform=matrix(.017,-0,0,-.017,75.639,15.95)><use xlink:href=#x2264/></g><g transform=matrix(.017,-0,0,-.017,90.343,15.95)><use xlink:href=#x32/></g> <g transform=matrix(.012,-0,0,-.012,98.512,7.8)><use xlink:href=#x1D441/></g> <g transform=matrix(.017,-0,0,-.017,113.312,15.95)><use xlink:href=#x2212/></g><g transform=matrix(.017,-0,0,-.017,127.064,15.95)><use xlink:href=#x31/></g> </svg> is <svg style=vertical-align:-3.50803pt;width:358.1875px; id=M15 height=20.2125 version=1.1 viewBox=0 0 358.1875 20.2125 width=358.1875 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,15.775)><use xlink:href=#x1D442/></g><g transform=matrix(.017,-0,0,-.017,13.118,15.775)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,18.999,15.775)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,24.881,15.775)><use xlink:href=#x1D441/></g> <g transform=matrix(.012,-0,0,-.012,39.738,7.613)><use xlink:href=#x33/></g> <g transform=matrix(.017,-0,0,-.017,49.85,15.775)><use xlink:href=#x2B/></g><g transform=matrix(.017,-0,0,-.017,63.602,15.775)><use xlink:href=#x1D440/></g><g transform=matrix(.017,-0,0,-.017,84.731,15.775)><use xlink:href=#x22C5/></g><g transform=matrix(.017,-0,0,-.017,92.381,15.775)><use xlink:href=#x1D441/></g> <g transform=matrix(.012,-0,0,-.012,107.25,7.613)><use xlink:href=#x32/></g> <g transform=matrix(.017,-0,0,-.017,117.362,15.775)><use xlink:href=#x22C5/></g><g transform=matrix(.017,-0,0,-.017,125.012,15.775)><use xlink:href=#x6C/></g><g transform=matrix(.017,-0,0,-.017,129.313,15.775)><use xlink:href=#x6F/></g><g transform=matrix(.017,-0,0,-.017,137.982,15.775)><use xlink:href=#x67/></g><g transform=matrix(.017,-0,0,-.017,145.937,15.775)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,151.819,15.775)><use xlink:href=#x1D441/></g><g transform=matrix(.017,-0,0,-.017,166.676,15.775)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,172.557,15.775)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,182.213,15.775)><use xlink:href=#x22C5/></g><g transform=matrix(.017,-0,0,-.017,189.862,15.775)><use xlink:href=#x4D/></g><g transform=matrix(.017,-0,0,-.017,204.464,15.775)><use xlink:href=#x75/></g><g transform=matrix(.017,-0,0,-.017,213.558,15.775)><use xlink:href=#x6C/></g><g transform=matrix(.017,-0,0,-.017,217.672,15.775)><use xlink:href=#x74/></g><g transform=matrix(.017,-0,0,-.017,222.925,15.775)><use xlink:href=#x69/></g><g transform=matrix(.017,-0,0,-.017,227.293,15.775)><use xlink:href=#x70/></g><g transform=matrix(.017,-0,0,-.017,236.082,15.775)><use xlink:href=#x6C/></g><g transform=matrix(.017,-0,0,-.017,240.382,15.775)><use xlink:href=#x69/></g><g transform=matrix(.017,-0,0,-.017,244.938,15.775)><use xlink:href=#x63/></g><g transform=matrix(.017,-0,0,-.017,252.163,15.775)><use xlink:href=#x61/></g><g transform=matrix(.017,-0,0,-.017,259.319,15.775)><use xlink:href=#x74/></g><g transform=matrix(.017,-0,0,-.017,264.572,15.775)><use xlink:href=#x69/></g><g transform=matrix(.017,-0,0,-.017,269.127,15.775)><use xlink:href=#x6F/></g><g transform=matrix(.017,-0,0,-.017,277.61,15.775)><use xlink:href=#x6E/></g><g transform=matrix(.017,-0,0,-.017,286.908,15.775)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,292.79,15.775)><use xlink:href=#x1D441/></g><g transform=matrix(.017,-0,0,-.017,307.647,15.775)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,317.285,15.775)><use xlink:href=#x2B/></g><g transform=matrix(.017,-0,0,-.017,331.037,15.775)><use xlink:href=#x1D441/></g> <g transform=matrix(.012,-0,0,-.012,345.912,7.613)><use xlink:href=#x34/></g> <g transform=matrix(.017,-0,0,-.017,352.25,15.775)><use xlink:href=#x29/></g> </svg>. After preprocessing, we can actually compute binomial coefficients modulo any <svg style=vertical-align:-0.0pt;width:16.575001px; id=M16 height=16.012501 version=1.1 viewBox=0 0 16.575001 16.012501 width=16.575001 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,15.95)><use xlink:href=#x32/></g> <g transform=matrix(.012,-0,0,-.012,8.225,7.8)><path id=x1D445 d=M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z /></g> </svg> with <svg style=vertical-align:-0.546pt;width:45.349998px; id=M17 height=11.85 version=1.1 viewBox=0 0 45.349998 11.85 width=45.349998 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,11.113)><use xlink:href=#x1D445/></g><g transform=matrix(.017,-0,0,-.017,15.718,11.113)><use xlink:href=#x2264/></g><g transform=matrix(.017,-0,0,-.017,30.422,11.113)><use xlink:href=#x1D441/></g> </svg>. For larger values of <svg style=vertical-align:-0.0pt;width:10.9375px; id=M18 height=11.175 version=1.1 viewBox=0 0 10.9375 11.175 width=10.9375 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,11.113)><use xlink:href=#x1D443/></g> </svg> and <svg style=vertical-align:-2.2932pt;width:13.175px; id=M19 height=14.3125 version=1.1 viewBox=0 0 13.175 14.3125 width=13.175 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,11.4)><use xlink:href=#x1D444/></g> </svg>, variations of Lucas’ theorem must be used first in order to reduce the computation to the evaluation of multiple <svg style=vertical-align:-3.50803pt;width:80.199997px; id=M20 height=16.6 version=1.1 viewBox=0 0 80.199997 16.6 width=80.199997 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.162)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,5.944,12.162)><use xlink:href=#x1D442/></g><g transform=matrix(.017,-0,0,-.017,18.999,12.162)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,24.881,12.162)><use xlink:href=#x6C/></g><g transform=matrix(.017,-0,0,-.017,29.181,12.162)><use xlink:href=#x6F/></g><g transform=matrix(.017,-0,0,-.017,37.851,12.162)><use xlink:href=#x67/></g><g transform=matrix(.017,-0,0,-.017,45.806,12.162)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,51.688,12.162)><use xlink:href=#x1D443/></g><g transform=matrix(.017,-0,0,-.017,62.499,12.162)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,68.38,12.162)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,74.262,12.162)><use xlink:href=#x29/></g> </svg> binomial coefficients <svg style=vertical-align:-2.3205pt;width:61.987499px; id=M21 height=23.612499 version=1.1 viewBox=0 0 61.987499 23.612499 width=61.987499 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,20.662)><use xlink:href=#x1D436/></g><g transform=matrix(.017,-0,0,-.017,12.047,20.662)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,17.928,20.662)><use xlink:href=#x1D443/></g> <g transform=matrix(.008,-0,0,-.008,28.738,6.8)><path id=x2032 d=M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z /></g> <g transform=matrix(.017,-0,0,-.017,32.512,20.662)><use xlink:href=#x2C/></g><g transform=matrix(.017,-0,0,-.017,39.21,20.662)><use xlink:href=#x1D444/></g> <g transform=matrix(.008,-0,0,-.008,52.275,6.8)><use xlink:href=#x2032/></g> <g transform=matrix(.017,-0,0,-.017,56.037,20.662)><use xlink:href=#x29/></g> </svg> (or restricted types of factorials <svg style=vertical-align:-0.1638pt;width:19.4px; id=M22 height=20.924999 version=1.1 viewBox=0 0 19.4 20.924999 width=19.4 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,20.662)><use xlink:href=#x1D443/></g> <g transform=matrix(.008,-0,0,-.008,10.875,6.8)><use xlink:href=#x2032/></g> <g transform=matrix(.017,-0,0,-.017,14.638,20.662)><path id=x21 d=M137 665q52 0 52 -30q0 -23 -7 -79q-10 -88 -29 -359q-4 -5 -17.5 -5t-16.5 5q-26 332 -29 359q-8 64 -8 79q0 30 55 30zM136 -12q-24 0 -39.5 16t-15.5 43q0 25 16 41.5t40 16.5q25 0 41 -16.5t16 -41.5q0 -27 -16 -43t-42 -16z /></g> </svg>) modulo <svg style=vertical-align:-0.0pt;width:19.3125px; id=M23 height=16.012501 version=1.1 viewBox=0 0 19.3125 16.012501 width=19.3125 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,15.95)><use xlink:href=#x32/></g> <g transform=matrix(.012,-0,0,-.012,8.225,7.8)><use xlink:href=#x1D441/></g> </svg> with <svg style=vertical-align:-2.2932pt;width:142.825px; id=M24 height=23.5875 version=1.1 viewBox=0 0 142.825 23.5875 width=142.825 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,20.662)><use xlink:href=#x30/></g><g transform=matrix(.017,-0,0,-.017,12.931,20.662)><use xlink:href=#x2264/></g><g transform=matrix(.017,-0,0,-.017,27.634,20.662)><use xlink:href=#x1D444/></g> <g transform=matrix(.008,-0,0,-.008,40.7,6.8)><use xlink:href=#x2032/></g> <g transform=matrix(.017,-0,0,-.017,49.188,20.662)><use xlink:href=#x2264/></g><g transform=matrix(.017,-0,0,-.017,63.874,20.662)><use xlink:href=#x1D443/></g> <g transform=matrix(.008,-0,0,-.008,74.7,6.8)><use xlink:href=#x2032/></g> <g transform=matrix(.017,-0,0,-.017,83.188,20.662)><use xlink:href=#x2264/></g><g transform=matrix(.017,-0,0,-.017,97.891,20.662)><use xlink:href=#x32/></g> <g transform=matrix(.012,-0,0,-.012,106.05,12.5)><use xlink:href=#x1D441/></g> <g transform=matrix(.017,-0,0,-.017,120.85,20.662)><use xlink:href=#x2212/></g><g transform=matrix(.017,-0,0,-.017,134.602,20.662)><use xlink:href=#x31/></g> </svg>." @default.
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W2038846491 date "2013-01-01" @default.
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W2038846491 title "A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two" @default.
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