Analysis of Prime Reciprocal Sequences in Base 10

Analysis of Prime Reciprocal Sequences in Base 10 Sumanth Kumar Reddy Gangasani Introduction The motivation for this study is the determination of structural redundancy in the prime reciprocals in base 10 in a manner that parallels a similar study for binary prime reciprocals [1-3]. Prime reciprocal sequences (also called d-sequences or decimal sequences for base 10) have applications in coding and cryptography [4-10] and for generation of random sequences [11-13]. We show that there are several different kinds of structural relationships amongst the digits in reciprocal sequences which are best classified with respect to the digit in the least significant place of the prime. Also, the frequency of digit 0 exceeds that of every other digit when the entire set of prime reciprocal sequences is considered. Generating the Sequence The formula used to generate the digits of the prime reciprocal sequences is as below [6]: a i = ( l( 10 i mod p)) mod 10 where a i is the i th digit in the sequence, p is the prime number and the value of l is given as below l = 1 if, p mod 10 = 9 l = 3 if, p mod 10 = 3 l = 7 if, p mod 10 = 7 l = 9 if, p mod 10 = 1 Digit Frequencies The least significant digit of the prime can be 1, 3, 7 or 9. One would expect different results for each of these cases. Furthermore, for any of these there would be further structure given depending on what the second least significant digit is when the sequence is half length. 1 Our analysis shows that Kak’s conjecture[6] that there are more 0s than 1s in most binary prime reciprocal sequences holds not only for binary and ternary prime reciprocal sequences[14] but also for decimal prime reciprocal sequences. Figure 1. Frequencies of digits 0 through 9 for prime reciprocal sequences The results of Figure 1 are based on prime reciprocal sequences for primes up to 999983. We expect that the excess of 0s over the other digits will persist as the values of primes are increased further. Structure in the Decimal Reciprocal Sequences The structure in the decimal expansion of various prime reciprocals becomes clear when we consider the primes with specific ending. The four cases, with the endings of 1, 3, 7, and 9, are given below. 2 T otal Fr e qu en cy 21 68 00 0 000 21 69 00 0 000 21 70 00 0 000 21 71 00 0 000 21 72 00 0 000 21 73 00 0 000 21 74 00 0 000 21 75 00 0 000 21 76 00 0 000 21 77 00 0 000 di gi t fr eque ncy 0 1 2 3 4 5 6 7 8 9 Sequence when primes end in 1: Full length sequences ending in 1 are observed to have equal number of each digit in every sequence. In any half length sequence ending in 1, when the second most significant digit is even, one would see that the frequency of 0 is equal to the frequency of 9 in every sequence, and maximum in many of the sequences. Digits 1, 2, 4, 5, 7 and 8 have the same frequency in each sequence and have either maximum or minimum frequency. The frequency of 3 is also seen to be the same as the frequency of 6 in every such sequence, and minimum in some sequences where 0s and 9s do not have maximum frequency. Table 1: Frequency distribution in the prime reciprocal sequences to the base-10 when the primes end in 1 and the 2 nd least significant digit is even. Prime No 0s 1s 2s 3s 4s 5s 6s 7s 8s 9s 601 35 28 28 31 28 28 31 28 28 35 3001 164 146 146 148 146 146 148 146 146 164 84401 4231 4251 4251 4116 4251 4251 4116 4251 4251 4231 473201 23737 23692 23692 23487 23692 23692 23487 23692 23692 23737 965801 48395 48350 48350 48005 48350 48350 48005 48350 48350 48395 6121 325 300 300 305 300 300 305 300 300 325 17321 881 869 869 842 869 869 842 869 869 881 317921 15925 15933 15933 15756 15933 15933 15756 15933 15933 15925 342521 17177 17164 17164 16961 17164 17164 16961 17164 17164 17177 940721 47144 47079 47079 46799 47079 47079 46799 47079 47079 47144 5441 276 277 277 253 277 277 253 277 277 276 22441 1157 1107 1107 1132 1107 1107 1132 1107 1107 1157 166841 8405 8337 8337 8294 8337 8337 8294 8337 8337 8405 394241 19795 19736 19736 19557 19736 19736 19557 19736 19736 19795 924641 46345 46269 46269 46008 46269 46269 46008 46269 46269 46345 761 40 39 39 33 39 39 33 39 39 40 73361 3687 3683 3683 3604 3683 3683 3604 3683 3683 3687 104761 5296 5217 5217 5243 5217 5217 5243 5217 5217 5296 371561 18605 18626 18626 18407 18626 18626 18407 18626 18626 18605 899161 45102 44920 44920 44928 44920 44920 44928 44920 44920 45102 881 49 44 44 39 44 44 39 44 44 49 5281 282 257 257 267 257 257 267 257 257 282 42281 2117 2133 2133 2054 2133 2133 2054 2133 2133 2117 309481 15561 15449 15449 15462 15449 15449 15462 15449 15449 15561 989081 49519 49508 49508 49227 49508 49508 49227 49508 49508 49519 3 In any half length sequence ending in 1, when the second most significant digit is odd, one would see that the sum of the frequencies of complementary digits is equal to the integral part of p /10. Number of 1s, 5s and 6s are equal and the number of 3s, 4s and 8s are also equal. It is also seen that either 0s or 2s have the maximum frequency and consequently either 7s or 9s have minimum frequency. Table 2: Frequency distribution in the prime reciprocal sequences to the base-10 when the primes end in 1 and the 2 nd least significant digit is odd. Prime No 0s 1s 2s 3s 4s 5 s 6s 7s 8s 9s 911 58 47 50 44 44 4 7 47 41 44 33 3511 185 181 192 170 170 1 81 181 159 170 166 33311 1771 1671 1682 1660 1660 1671 1671 1649 1660 1560 388111 19510 19452 19545 19359 19359 1 9452 19452 19266 19359 19301 997511 50165 49925 50024 49826 49826 4 9925 49925 49727 49826 49586 631 33 34 39 29 29 3 4 34 24 29 30 5231 290 266 275 257 257 2 66 266 248 257 233 77431 3905 3904 3969 3839 3839 3904 3904 3774 3839 3838 454031 22950 22738 22811 22665 22665 2 2738 22738 22592 22665 22453 911831 46114 45646 45755 45537 45537 4 5646 45646 45428 45537 45069 151 8 9 12 6 6 9 9 3 6 7 2351 144 120 125 115 115 1 20 120 110 115 91 57751 2909 2911 2958 2864 2864 2911 2911 2817 2864 2866 288551 14666 14459 14522 14396 14396 1 4459 14459 14333 14396 14189 998951 50399 49997 50096 49898 49898 4 9997 49997 49799 49898 49496 1471 76 78 87 69 69 7 8 78 60 69 71 23071 1185 1164 1185 1143 1143 1164 1164 1122 1143 1122 76871 3934 3856 3881 3831 3831 3856 3856 3806 3831 3753 597671 30285 29918 29987 29849 29849 2 9918 29918 29780 29849 29482 996271 49978 49891 50046 49736 49736 4 9891 49891 49581 49736 49649 991 51 53 60 46 46 5 3 53 39 46 48 9391 472 482 507 457 457 4 82 482 432 457 467 27791 1476 1397 1412 1382 1382 1397 1397 1367 1382 1303 347591 17826 17397 17432 17362 17362 1 7397 17397 17327 17362 16933 878191 44069 43998 44175 43821 43821 4 3998 43998 43644 43821 43750 Sequence when primes end in 3: In full length sequences, number of 3s and 6s is 1 more than the number of any other digit and all other digits have the same frequency. 4 In a half length sequence that ends with 3, if the 2 nd least significant digit is even, the sum of the frequencies of the complementary digits is equal to the integral part of p /10. Number of 0s and 9s is always seen to be equal. Digits 1, 4 and 7 are seen to be equally frequent and since the complementary digits are 8, 5 and 2 respectively, number of 2s, 5s and 8s is also equal. The digit 3 has maximum frequency and as expected the complementary digit 6 has minimum frequency. Table 3: Frequency distribution in the prime reciprocal sequences to the base-10 when the prime ends in 3 and the 2 nd least significant digit is even. Prime No 0s 1s 2s 3 s 4s 5s 6s 7s 8s 9s 2203 110 101 119 127 101 119 94 101 119 110 5003 250 251 249 272 251 249 229 251 249 250 64403 3220 3245 3195 3308 3245 3195 3133 3245 3195 3220 431603 21580 21593 21567 21869 21593 21567 21292 21593 21567 21580 996803 49840 49936 49744 50145 49936 49744 49536 49936 49744 49840 523 26 24 28 36 24 28 17 24 28 26 5923 296 275 317 328 275 317 265 275 317 296 92723 4636 4628 4644 4733 4628 4644 4540 4628 4644 4636 354323 17716 17718 17714 17932 17718 17714 17501 17718 17714 17716 954323 47716 47810 47622 48110 47810 47622 47323 47810 47622 47716 443 22 23 21 29 23 21 16 23 21 22 7643 382 396 368 412 396 368 353 396 368 382 49043 2452 2453 2451 2522 2453 2451 2383 2453 2451 2452 382843 19142 19025 19259 19387 19025 19259 18898 19025 19259 19142 844243 42212 42025 42399 42542 42025 42399 41883 42025 42399 42212 563 28 32 24 38 32 24 19 32 24 28 8963 448 460 436 480 460 436 417 460 436 448 19763 988 996 980 1042 996 980 935 996 980 988 498163 24908 24711 25105 25239 24711 25105 24578 24711 25105 24908 950363 47518 47467 47569 47910 47467 47569 47127 47467 47569 47518 683 34 33 35 43 33 35 26 33 35 34 6883 344 323 365 379 323 365 310 323 365 344 27283 1364 1332 1396 1431 1332 1396 1298 1332 1396 1364 233083 11654 11537 11771 11848 11537 11771 11461 11537 11771 11654 985483 49274 48964 49584 49721 48964 49584 48828 48964 49584 49274 In a half length sequence that ends with 3, if the second least significant digit is odd, the complementary digits have equal frequencies. In most of the cases 1s and 8s or 2s and 7s are observed to have maximum frequency. In all these sequences, 3s and 6s or 4s and 5s have the minimum frequency. In other sequences, 0s and 9s have maximum frequency and 1s and 8s have the minimum frequency. 5 Table 4: Frequency distribution in the prime reciprocal sequences to the base-10 when the primes end in 3 and the 2 nd least significant digit is odd. Prime No 0s 1s 2s 3s 4 s 5s 6 s 7s 8 s 9s 5413 278 267 282 267 259 259 267 282 267 278 49613 2467 2 522 2495 2 453 2466 2 466 2453 2495 2522 2467 89213 4458 4 498 4493 4 426 4428 4 428 4426 4493 4498 4458 235013 11723 11855 11800 11674 11701 11701 11674 11800 11855 11723 914813 45679 45939 45816 45604 45665 45665 45604 45816 45939 45679 3533 178 180 183 172 170 170 172 183 180 178 18133 925 890 927 905 886 886 905 927 890 925 48733 2474 2 398 2473 2 438 2400 2 400 2438 2473 2398 2474 266333 13286 13422 13361 13242 13272 13272 13242 13361 13422 13286 986933 49373 49390 49443 49277 49250 49250 49277 49443 49390 49373 653 32 36 35 30 3 0 30 3 0 35 3 6 32 6053 302 312 311 294 294 294 294 311 312 302 61253 3071 3 078 3095 3 039 3030 3 030 3039 3095 3078 3071 391453 19673 19468 19669 19577 19476 19476 19577 19669 19468 19673 984853 49348 49173 49384 49207 49101 49101 49207 49384 49173 49348 373 20 18 21 18 1 6 16 1 8 21 1 8 20 6173 309 315 316 302 301 301 302 316 315 309 70573 3562 3 490 3557 3 534 3500 3 500 3534 3557 3490 3562 228773 11444 11479 11490 11393 11387 11387 11393 11490 11479 11444 982973 49149 49260 49261 49037 49036 49036 49037 49261 49260 49149 293 14 17 16 13 1 3 13 1 3 16 1 7 14 8093 396 427 410 391 399 399 391 410 427 396 33493 1703 1 650 1707 1 671 1642 1 642 1671 1707 1650 1703 449693 22479 22563 22552 22412 22417 22417 22412 22552 22563 22479 993893 49724 49748 49807 49612 49582 49582 49612 49807 49748 49724 Sequence when primes end in 7: In full length sequences, number of 0s, 3s, 6s and 9s is 1 less than the number of other digits. In a half length sequence that ends with 7, if the second least significant digit is even, sum of the frequencies of complementary digits for 0s and 3s is equal to integral part of p /10 and 1 more than integral part of p /10 for others. The digits 1, 4 and 7 have equal frequencies and hence the complementary digits 2, 5 and 8 have the same frequency. It is also seen that 3 occurs maximum number of times in every sequence and 6 occurs minimum number of times in every such sequence. 6 Table 5: Frequency distribution in the prime reciprocal sequences to the base-10 when the primes end in 7 and the 2 nd least significant digit is even. Prime No 0s 1s 2s 3s 4s 5s 6s 7s 8s 9s 307 15 13 18 22 13 18 8 13 18 15 5507 275 285 266 300 285 266 250 285 266 2 75 17107 855 835 876 907 835 876 803 835 876 8 55 195907 9795 9 754 9837 9952 9754 9837 9638 9 754 9837 9795 953707 47685 47516 47855 48132 47516 47855 47238 47516 47855 47685 827 41 41 42 52 41 42 30 41 42 41 2027 101 103 100 116 103 100 86 103 100 101 23227 1161 1 132 1191 1222 1132 1191 1100 1 132 1191 1161 139627 6981 6 929 7034 7107 6929 7034 6855 6 929 7034 6981 963427 48171 48101 48242 48459 48101 48242 47883 48101 48242 48171 947 47 46 49 56 46 49 38 46 49 47 5147 257 262 253 281 262 253 233 262 253 2 57 33547 1677 1 637 1718 1758 1637 1718 1596 1 637 1718 1677 197347 9867 9 820 9915 10027 9820 9915 9707 9820 9915 9 867 995747 49787 49828 49747 50087 49828 49747 49487 49828 49747 49787 467 23 27 20 30 27 20 16 27 20 23 3067 153 141 166 176 141 166 130 141 166 1 53 25667 1283 1 300 1267 1334 1300 1267 1232 1 300 1267 1283 313267 15663 15627 15700 15839 15627 15700 15487 15627 15700 15663 992867 49643 49652 49635 50035 49652 49635 49251 49652 49635 49643 787 39 35 44 51 35 44 27 35 44 39 5387 269 272 267 301 272 267 237 272 267 2 69 16187 809 819 800 861 819 800 757 819 800 8 09 330587 16529 16564 16495 16706 16564 16495 16352 16564 16495 16529 995987 49799 49878 49721 50076 49878 49721 49522 49878 49721 49799 In a half length sequences that end in 7, if the second least significant digit is odd, the complementary digits have equal frequencies. In most of the cases 1s and 8s or 2s and 7s are observed to have maximum frequency. In all these sequences, 3s and 6s or 4s and 5s have the minimum frequency. In other sequences, 0s and 9s have maximum frequency and 1s and 8s have the minimum frequency. This structure of the digit frequencies of half length sequences the end in 7, when compared to the structure of the digit frequencies of half length sequences that end with the complementary digit 3, shows a lot of similarity. 7 Table 6: Frequency distribution in the prime reciprocal sequences to the base-10 when the primes end in 7 and the 2 nd least significant digit is odd. Sequence when primes end in 9: In full length sequences, number of 0s is equal to the number of 9s which is one less than the frequency of any other digit. All other digits have the same frequency. In a half length sequence that ends in 9, when the 2 nd least significant digit is even, complementary digits are equal. It is also seen that digits 1, 2 and 4 and hence 5, 7 and 8 have the same frequency and are mostly minimum in number. 0s and 9s are mostly highest in number. In few of the sequences, 1s, 2s, 4s, 5s, 7s and 8s have highest frequency and 3s and 6s have the least frequency. 8 Prime No 0s 1s 2s 3s 4s 5s 6s 7s 8s 9s 2917 152 140 153 145 139 139 145 153 140 152 14717 739 742 749 726 723 723 726 749 742 739 74317 3769 3658 3765 3720 3667 3667 3720 3765 3658 3769 243517 12222 12155 12248 12150 12104 12104 12150 12248 12155 1 2222 999917 49975 50129 50088 49883 49904 49904 49883 50088 50129 4 9975 2437 125 121 128 119 116 116 119 128 121 125 51637 2620 2542 2619 2583 2545 2545 2583 2619 2542 2620 92237 4611 4641 4640 4583 4584 4584 4583 4640 4641 4611 209837 10519 10496 10551 10460 10433 10433 10460 10551 10496 1 0519 997037 49840 49976 49953 49739 49751 49751 49739 49953 49976 4 9840 557 26 32 29 25 27 27 25 29 32 26 4157 208 214 215 201 201 201 201 215 214 208 33757 1721 1650 1717 1692 1659 1659 1692 1717 1650 1721 179957 8965 9090 9025 8938 8971 8971 8938 9025 9090 8965 977357 48926 48871 48988 48806 48748 48748 48806 48988 48871 4 8926 877 46 43 48 42 40 40 42 48 43 46 30677 1536 1549 1554 1516 1514 1514 1516 1554 1549 1536 15077 756 762 767 743 741 741 743 767 762 756 248477 12442 12449 12486 12380 12362 12362 12380 12486 12449 1 2442 985277 49247 49391 49358 49153 49170 49170 49153 49358 49391 4 9247 197 9 12 11 8 9 9 8 11 12 9 4597 238 224 241 227 219 219 227 241 224 238 18397 932 916 941 911 899 899 911 941 916 932 795997 39942 39653 39938 39804 39662 39662 39804 39938 39653 3 9942 989797 49567 49456 49611 49446 49369 49369 49446 49611 49456 4 9567 Table 7: Frequency distribution in the prime reciprocal sequences to the base-10 when the primes end in 9 and the 2 nd least significant digit is even. Prime No 0s 1s 2s 3s 4s 5s 6s 7s 8s 9s 409 24 19 19 2 1 19 19 21 19 19 24 3209 166 162 162 150 162 162 150 162 162 166 17609 891 892 892 835 892 892 835 892 892 891 194809 9832 9705 9705 9755 9705 9705 9755 9705 9705 9832 974009 48822 48750 48750 48430 48750 48750 48430 48750 48750 48822 929 50 47 47 4 1 47 47 41 47 47 50 4129 220 201 201 209 201 201 209 201 201 220 24329 1225 1226 1226 1179 1226 1226 1179 1226 1226 1225 254729 12790 12748 12748 12648 12748 12748 12648 12748 12748 12790 996529 49988 49766 49766 49846 49766 49766 49846 49766 49766 49988 1049 57 52 52 4 9 52 52 49 52 52 57 48449 2438 2441 2441 2351 2441 2441 2351 2441 2441 2438 56249 2817 2829 2829 2758 2829 2829 2758 2829 2829 2817 304849 15346 15196 15196 15278 15196 15196 15278 15196 15196 15346 996649 50044 49726 49726 49940 49726 49726 49940 49726 49726 50044 569 30 30 30 2 2 30 30 22 30 30 30 8969 459 452 452 427 452 452 427 452 452 459 46769 2347 2353 2353 2286 2353 2353 2286 2353 2353 2347 230369 11584 11523 11523 11439 11523 11523 11439 11523 11523 11584 993169 49833 49616 49616 49611 49616 49616 49611 49616 49616 49833 2089 113 101 101 106 101 101 106 101 101 113 30689 1543 1544 1544 1497 1544 1544 1497 1544 1544 1543 76289 3850 3820 3820 3762 3820 3820 3762 3820 3820 3850 602489 30248 30122 30122 30008 30122 30122 30008 30122 30122 30248 990889 49795 49447 49447 49586 49447 49447 49586 49447 49447 49795 In a half length sequence that ends with 9, if the second least significant digit is odd, sum of the frequencies of complementary digits for 0s is equal to integral part of p /10 and 1 more than integral part of p /10 for others. The digits 1, 5 and 6 have equal frequencies and hence the complementary digits 3, 4 and 8 have the same frequency. It is also seen that 0s or 2s are most frequent while 7s or 9s are least frequent in all such sequences. As seen in case of 7 and 3, the digit frequency structure of half-length sequences that end in 9 and those that end in 1 have similar structure. 9 Table 8: Frequency distribution in the prime reciprocal sequences to the base-10 when the primes end in 9 and the 2 nd least significant digit is odd. Prime No 0s 1s 2s 3s 4s 5s 6s 7s 8s 9s 919 49 49 55 4 3 43 49 49 37 43 42 5519 316 280 288 272 272 280 280 264 272 235 23719 1209 1196 1216 1176 1176 1196 1196 1156 1176 1162 201119 10310 10078 10122 10034 10034 10078 10078 9990 10034 9801 994319 50080 49778 49902 49654 49654 49778 49778 49530 49654 49351 839 56 43 45 4 1 41 43 43 39 41 27 4639 243 239 253 225 225 239 239 211 225 220 34439 1812 1730 1746 1714 1714 1730 1730 1698 1714 1631 348239 17676 17436 17484 17388 17388 17436 17436 17340 17388 17147 994039 49752 49816 50044 49588 49588 49816 49816 49360 49588 49651 359 25 19 21 1 7 17 19 19 15 17 10 1759 93 92 100 84 8 4 92 92 76 84 82 23159 1233 1163 1173 1153 1153 1163 1163 1143 1153 1082 346559 17586 17354 17406 17302 17302 17354 17354 17250 17302 17069 999959 50546 50043 50133 49953 49953 50043 50043 49863 49953 49449 479 34 25 27 2 3 23 25 25 21 23 13 3079 164 159 169 149 149 159 159 139 149 143 44279 2304 2223 2241 2205 2205 2223 2223 2187 2205 2123 193679 10007 9703 9741 9665 9665 9703 9703 9627 9665 9360 961879 48255 48190 48382 47998 47998 48190 48190 47806 47998 47932 599 40 31 33 2 9 29 31 31 27 29 19 6199 307 321 343 299 299 321 321 277 299 312 41399 2172 2079 2097 2061 2061 2079 2079 2043 2061 1967 445799 22651 22323 22389 22257 22257 22323 22323 22191 22257 21928 989999 50242 49530 49590 49470 49470 49530 49530 49410 49470 48757 In addition to the structural redundancy seen in all these sequences, the prime numbers also are seen to have a definite structure. For each combination of the least significant digit and the next least significant digit, the third least significant digit is observed to be alternating between even and odd values when the second least significant digit is either even or odd. This property can be seen in every table above. 1 Conclusions We have shown several interesting structural properties of non-maximum length decimal sequences of prime reciprocals. We have classified these properties in relation to the digit in the least significant place of the prime. We have also observed, based on the experiment with primes less than a million, that the frequency of 0 exceeds that of other digits, in accordance with the corresponding result on binary prime reciprocal sequences. If prime reciprocal sequences are used in cryptography applications, as has been proposed earlier, then these structural properties must be taken into consideration. References 1. S. Kak, A structural redundancy in d-sequences. IEEE Transactions on Computers, vol. C-32, pp. 1069-1070, 1983. 2. S.K.R. Gangasani, Testing Kak’s Conjecture on Binary Reciprocal of Primes and 3. S. Kak, Prime reciprocals and the Euler zeta function. 2007. http://www.cs.okstate.edu/~subhashk/Prime.pdf 4. S. Kak and A. Chatterjee, On decimal sequences. IEEE Transactions on Information Theory, vol. IT-27, pp. 647 – 652, 1981. 5. S. Kak, Encryption and error-correction coding using D sequences. IEEE Transactions on Computers, vol. C-34, pp. 803-809, 1985. 6. S. Kak, New results on d-sequences. Electronics Letters, vol. 23, p. 617, 1987. 7. D. Mandelbaum, On subsequences of arithmetic sequences. IEEE Transactions on Computers, vol. C-37, pp 1314-1315, 1988. 8. S. Kak, A new method for coin flipping by telephone. Cryptologia, vol. 13, pp. 73-78, 1989. 9. N. Mandhani and S. Kak, Watermarking using decimal sequences. Cryptologia, vol 29, pp. 50-58, 2005; arXiv: cs.CR/0602003 1 10. A.K. Parthasarathy and S. Kak, An improved method of content based image watermarking, IEEE Trans on Broadcasting, vol. 53, pp. 468-479, 2007. 11. A. Parakh , A d-sequence based recursive random number generator, 2006; arXiv: cs/0603029 12. S.K.R. Gangasani, Testing d-sequences for their randomness, 2007; arXiv: 0710.3779 13. S.B. Thippireddy and S. Chalasani, Period of d-sequence based random number 14. S.K.R Gangasani, Testing Kak's Conjecture on Binary Reciprocal of Primes and 1