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# Introduction to Number Theory MATH 3240

UCONN

GPA 3.97

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This 4 page Study Guide was uploaded by Mary Veum on Thursday September 17, 2015. The Study Guide belongs to MATH 3240 at University of Connecticut taught by Keith Conrad in Fall. Since its upload, it has received 37 views. For similar materials see /class/205818/math-3240-university-of-connecticut in Mathematics (M) at University of Connecticut.

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Date Created: 09/17/15

DECIMAL DATA The rst two tables below give the decimal expansion of 1b for 2 S b S 90 Some expansions do not have their periodic part at the beginning eg7 144 02 102272727271 1 i has an initial 02 Which is not in the periodic parti Those 1b having a purely periodic decimal expansion that is7 it repeats right from the start are printed in bold A nite decimal expansion could be Viewed as an expansion With periodic part 0 eg7 116 10625 10625 i Fraction Decimal Expansion 12 5 13 g 14 25 15 20 16 13 17 1142857 18 125 19 1 110 1 111 E 112 USE 113 076923 114 10714528 115 03 116 1 117 10588235294117647 118 03 119 1052631578947368421 120 1 121 047619 122 1 i 123 10434782608695652173913 124 7 125 04 126 10384615 127 1 128 103571428 129 10344827586206896551724137931 130 DE 131 03225806451612 132 103125 133 if 134 102941176470588235 135 10285714 136 1027 137 1W 138 10263157894736842105 139 140 1025 DECIMAL DATA Fraction Decimal Expansion 141 02439 142 0238095 143 023255813953488372093 144 02 145 1 7 146 02173913043478260869565 147 0212 6595 446 U 51003 9 I 404255 1914893617 148 1 149 020408163265306122448979591836734693 551 150 02 151 0196078431372549 152 01923076 153 0188679245283 154 0 155 0 156 017857142 157 017543859649122807 158 01724137931034482758620689655 159 i0169491525423728813559322033898305084745762711864406779661 160 1 6 161 1 WAN quot E39I 1901131141 RAIN hllhhh Imw I 162 0161290322580645 163 015873015873 164 015625 165 0153846 166 1 i 167 i014925373134328358208955223880597 168 014705882352941176 169 0144927536231884057971 170 1 171 01408450704225352112676056338028169 172 1 7 173 01369863 174 0 175 1 7 176 01315789473684210526 177 1 12987 178 0128205 179 0126582278481 180 0125 181 012345679 182 1 183 01204819277108433734939759036144578313253 184 01190476 185 01176470588235294 186 0116279069767441860465 187 0114942528735632183908045977 188 1 i 189 01123595505617977528089887640449438202247191 01 190 DECIMAL DATA 3 The next table indicates the period length for the decimal expansion of 1p when p is a prime up to 1137 other than 2 and 5 7 11 13 17 19 23 29 31 37 41 43 47 53 2 6 16 18 22 28 15 3 5 21 46 13 61 67 71 73 79 83 89 97 101 103 107 109 113 33 35 8 13 41 44 96 4 34 53 108 112 p length1p H on p 59 length1p 5 00 on O So far we have focused on unit fractions 1bi Now we will compare the decimal expansions of all reduced proper fractions with a common denominator like 177 277 37i i i 67 In the tables below7 only purely periodic decimals are listed Fraction Decimal Fraction Decimal 13 7 23 6 17 1142857 27 1285714 37 1428571 47 1571428 57 1714285 67 1857142 19 1 29 2 49 4 59 5 79 7 89 8 111 1 211 1 311 1 411 1 511 lg 611 711 E 811 1 911 lg 1011 m 113 076923 213 153846 313 230769 413 307692 513 384615 613 461538 713 538461 813 615384 913 692307 1013 1769230 1113 1846153 1213 1923076 117 i0588235294117647 217 i1176470588235294 317 i1764705882352941 417 2352941176470588 517 i2941176470588235 617 3529411764705882 717 i4117647058823529 817 i4705882352941176 917 5294117647058823 1017 5882352941176470 1117 6470588235294117 1217 i7058823529411764 1317 i7647058823529411 1417 8235294117647058 1517 8823529411764705 1617 9411764705882352 119 052631578947368421 219 1105263157894736842 319 1157894736842105263 419 1210526315789473684 519 1263157894736842105 619 1315789473684210526 719 1368421052631578947 819 1421052631578947368 919 1473684210526315789 1019 1526315789473684210 1119 578947368421052631 1219 1631578947368421052 1319 1684210526315789473 1419 1736842105263157894 1519 1789473684210526315 1619 1842105263157894736 1719 1894736842105263157 1819 1947368421052631578 DECIMAL DATA Fraction Decimal Fraction Decimal 121 047619 221 095238 421 190476 521 238095 821 380952 1021 1476190 1121 1523809 1321 1619047 1621 1761904 1721 1809523 1921 1904761 2021 1952380 123 10434782608695652173913 223 10869565217391304347826 323 11304347826086956521739 423 11739130434782608695652 523 12173913043478260869565 623 12608695652173913043478 723 13043478260869565217391 823 13478260869565217391304 923 13913043478260869565217 1023 14347826086956521739130 1123 14782608695652173913043 1223 15217391304347826086956 1323 15652173913043478260869 1423 16086956521739130434782 1523 16521739130434782608695 1623 16956521739130434782608 1723 17391304347826086956521 1823 17826086956521739130434 1923 18260869565217391304347 2023 18695652173913043478260 2123 19130434782608695652173 2223 19565217391304347826086 127 1W 227 M 427 W 527 E 727 827 M 1027 W 1127 W 1327 m 1427 1 1627 W 1727 W 1927 m 2027 W 2227 m 2327 2527 2627 W

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