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# large prime

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The number 2^4253 -1 is a very large prime find how many digits it would take to write the number in decimal form. ( Hint: Log 2 is an interesting irrational number It's the power of 10 that gives 2, in the same way, that 3 is the power of 10 that gives 100. Your calculator can calculate log 2)

Oct 15, 2020

#1
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2^4253 - 1 = 190 797007 524439 073807 468042 969529 173669 356994 749940 177394 741882 673528 979787 005053 706368 049835 514900 244303 495954 950709 725762 186311 224148 828811 920216 904542 206960 744666 169364 221195 289538 436845 390250 168663 932838 805192 055137 154390 912666 527533 007309 292687 539092 257043 362517 857366 624699 975402 375462 954490 293259 233303 137330 643531 556539 739921 926201 438606 439020 075174 723029 056838 272505 051571 967594 608350 063404 495977 660656 269020 823960 825567 012344 189908 927956 646011 998057 988548 630107 637380 993519 826582 389781 888135 705408 653045 219655 801758 081251 164080 554609 057468 028203 308718 724654 081055 323215 860189 611391 296030 471108 443146 745671 967766 308925 858547 271507 311563 765171 008318 248647 110097 614890 313562 856541 784154 881743 146033 909602 737947 385055 355960 331855 614540 900081 456378 659068 370317 267696 980001 187750 995491 090350 108417 050917 991562 167972 281070 161305 972518 044872 048331 306383 715094 854938 415738 549894 606070 722584 737978 176686 422134 354526 989443 028353 644037 187375 385397 838259 511833 166416 134323 695660 367676 897722 287918 773420 968982 326089 026150 031515 424165 462111 337527 431154 890666 327374 921446 276833 564519 776797 633875 503548 665093 914556 482031 482248 883127 023777 039667 707976 559857 333357 013727 342079 099064 400455 741830 654320 379350 833236 245819 348824 064783 585692 924881 021978 332974 949906 122664 421376 034687 815350 484991 (1281 digits) is prime

Oct 15, 2020
#2
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Of course, you can find out the exact number of decimal digits this way:

2^4253 - 1 =4253 x log_10(2) =4253 x 0.3010299957 =1,280.2805715589 =1280 + 1(for the fraction)=1281 decimal digits. The subtraction of 1 makes no difference to the number of decimal digits.

Oct 15, 2020