2006-06-13 07:29:51 · 4 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
The range of consecutive prime numbers for a "strictly" 9-digit product begins at 463 (* 467 * 479 = 103569859) and ends with 991 (* 997 * 1009 = 996919243). Within this numerical range, there are NO consecutive primes whose product results in unique digits.
Even if you expand the definition of a 9-digit product to include those with a leading zero as a digit, you can also check all 8-digit products and consider the first digit to be a zero. The range then begins at 211 (* 223 * 227 = 010681031). Unfortunately, even with this expanded definition, there are no products with unique digits. (The solution 233*239*241=13420567 is incorrect because if expressed as nine digits, the product becomes 013420567 which repeats the 0 digit.)
For the morbidly curious, the full solution set follows:
211 x 223 x 227 = 10681031 -- 0x3 1x3 repeat
223 x 227 x 229 = 11592209 -- 0x2 1x2 2x2 9x2 repeat
227 x 229 x 233 = 12112039 -- 0x2 1x3 2x2 repeat
229 x 233 x 239 = 12752323 -- 2x3 3x2 repeat
233 x 239 x 241 = 13420567 -- 0x2 repeat
239 x 241 x 251 = 14457349 -- 4x3 repeat
241 x 251 x 257 = 15546187 -- 1x2 5x2 repeat
251 x 257 x 263 = 16965341 -- 1x2 6x2 repeat
257 x 263 x 269 = 18181979 -- 1x3 8x2 9x2 repeat
263 x 269 x 271 = 19172437 -- 1x2 7x2 repeat
269 x 271 x 277 = 20193023 -- 0x3 2x2 3x2 repeat
271 x 277 x 281 = 21093827 -- 0x2 2x2 repeat
277 x 281 x 283 = 22027871 -- 0x2 2x3 7x2 repeat
281 x 283 x 293 = 23300239 -- 0x3 2x2 3x3 repeat
283 x 293 x 307 = 25456133 -- 3x2 5x2 repeat
293 x 307 x 311 = 27974761 -- 7x3 repeat
307 x 311 x 313 = 29884301 -- 0x2 8x2 repeat
311 x 313 x 317 = 30857731 -- 0x2 3x2 7x2 repeat
313 x 317 x 331 = 32842151 -- 1x2 2x2 repeat
317 x 331 x 337 = 35360399 -- 0x2 3x3 9x2 repeat
331 x 337 x 347 = 38706809 -- 0x3 8x2 repeat
337 x 347 x 349 = 40811711 -- 0x2 1x4 repeat
347 x 349 x 353 = 42749359 -- 4x2 9x2 repeat
349 x 353 x 359 = 44227723 -- 2x3 4x2 7x2 repeat
353 x 359 x 367 = 46508809 -- 0x3 8x2 repeat
359 x 367 x 373 = 49143869 -- 4x2 9x2 repeat
367 x 373 x 379 = 51881689 -- 1x2 8x3 repeat
373 x 379 x 383 = 54143561 -- 1x2 4x2 5x2 repeat
379 x 383 x 389 = 56466073 -- 0x2 6x3 repeat
383 x 389 x 397 = 59147839 -- 9x2 repeat
389 x 397 x 401 = 61927633 -- 3x2 6x2 repeat
397 x 401 x 409 = 65111573 -- 1x3 5x2 repeat
401 x 409 x 419 = 68719771 -- 1x2 7x3 repeat
409 x 419 x 421 = 72147191 -- 1x3 7x2 repeat
419 x 421 x 431 = 76027969 -- 0x2 6x2 7x2 9x2 repeat
421 x 431 x 433 = 78568283 -- 8x3 repeat
431 x 433 x 439 = 81927497 -- 7x2 9x2 repeat
433 x 439 x 443 = 84208541 -- 0x2 4x2 8x2 repeat
439 x 443 x 449 = 87320173 -- 0x2 3x2 7x2 repeat
443 x 449 x 457 = 90900499 -- 0x4 9x4 repeat
449 x 457 x 461 = 94593973 -- 3x2 9x3 repeat
457 x 461 x 463 = 97543451 -- 4x2 5x2 repeat
461 x 463 x 467 = 99677881 -- 7x2 8x2 9x2 repeat
463 x 467 x 479 = 103569859 -- 5x2 9x2 repeat
467 x 479 x 487 = 108938491 -- 1x2 8x2 9x2 repeat
479 x 487 x 491 = 114537043 -- 1x2 3x2 4x2 repeat
487 x 491 x 499 = 119319383 -- 1x3 3x3 9x2 repeat
491 x 499 x 503 = 123239527 -- 2x3 3x2 repeat
499 x 503 x 509 = 127757473 -- 7x4 repeat
503 x 509 x 521 = 133390067 -- 0x2 3x3 repeat
509 x 521 x 523 = 138693847 -- 3x2 8x2 repeat
521 x 523 x 541 = 147413303 -- 1x2 3x3 4x2 repeat
523 x 541 x 547 = 154769821 -- 1x2 repeat
541 x 547 x 557 = 164831339 -- 1x2 3x3 repeat
547 x 557 x 563 = 171534277 -- 1x2 7x3 repeat
557 x 563 x 569 = 178433279 -- 3x2 7x2 repeat
563 x 569 x 571 = 182918137 -- 1x3 8x2 repeat
569 x 571 x 577 = 187466723 -- 6x2 7x2 repeat
571 x 577 x 587 = 193397129 -- 1x2 3x2 9x3 repeat
577 x 587 x 593 = 200848507 -- 0x3 8x2 repeat
587 x 593 x 599 = 208506509 -- 0x3 5x2 repeat
593 x 599 x 601 = 213479407 -- 4x2 7x2 repeat
599 x 601 x 607 = 218519393 -- 1x2 3x2 9x2 repeat
601 x 607 x 613 = 223626691 -- 2x3 6x3 repeat
607 x 613 x 617 = 229580147 -- 2x2 repeat
613 x 617 x 619 = 234118799 -- 1x2 9x2 repeat
617 x 619 x 631 = 240993413 -- 3x2 4x2 9x2 repeat
619 x 631 x 641 = 250367549 -- 5x2 repeat
631 x 641 x 643 = 260074853 -- 0x2 repeat
641 x 643 x 647 = 266669461 -- 6x5 repeat
643 x 647 x 653 = 271661713 -- 1x3 6x2 7x2 repeat
647 x 653 x 659 = 278421569 -- 2x2 repeat
653 x 659 x 661 = 284446147 -- 4x4 repeat
659 x 661 x 673 = 293158127 -- 1x2 2x2 repeat
661 x 673 x 677 = 301165481 -- 1x3 repeat
673 x 677 x 683 = 311189143 -- 1x4 3x2 repeat
677 x 683 x 691 = 319512181 -- 1x4 repeat
683 x 691 x 701 = 330839053 -- 0x2 3x4 repeat
691 x 701 x 709 = 343433219 -- 3x4 4x2 repeat
701 x 709 x 719 = 357349471 -- 3x2 4x2 7x2 repeat
709 x 719 x 727 = 370603517 -- 0x2 3x2 7x2 repeat
719 x 727 x 733 = 383148629 -- 3x2 8x2 repeat
727 x 733 x 739 = 393806449 -- 3x2 4x2 9x2 repeat
733 x 739 x 743 = 402473441 -- 4x4 repeat
739 x 743 x 751 = 412356827 -- 2x2 repeat
743 x 751 x 757 = 422400701 -- 0x3 2x2 4x2 repeat
751 x 757 x 761 = 432633827 -- 2x2 3x3 repeat
757 x 761 x 769 = 443003213 -- 0x2 3x3 4x2 repeat
761 x 769 x 773 = 452366557 -- 5x3 6x2 repeat
769 x 773 x 787 = 467821919 -- 1x2 9x2 repeat
773 x 787 x 797 = 484855747 -- 4x3 5x2 7x2 8x2 repeat
787 x 797 x 809 = 507436351 -- 3x2 5x2 repeat
797 x 809 x 811 = 522910903 -- 0x2 2x2 9x2 repeat
809 x 811 x 821 = 538657279 -- 5x2 7x2 repeat
811 x 821 x 823 = 547978913 -- 7x2 9x2 repeat
821 x 823 x 827 = 558789841 -- 5x2 8x3 repeat
823 x 827 x 829 = 564234809 -- 4x2 repeat
827 x 829 x 839 = 575204137 -- 5x2 7x2 repeat
829 x 839 x 853 = 593287943 -- 3x2 9x2 repeat
839 x 853 x 857 = 613326619 -- 1x2 3x2 6x3 repeat
853 x 857 x 859 = 627947039 -- 7x2 9x2 repeat
857 x 859 x 863 = 635308669 -- 3x2 6x3 repeat
859 x 863 x 877 = 650135009 -- 0x3 5x2 repeat
863 x 877 x 881 = 666785731 -- 6x3 7x2 repeat
877 x 881 x 883 = 682238471 -- 2x2 8x2 repeat
881 x 883 x 887 = 690017701 -- 0x3 1x2 7x2 repeat
883 x 887 x 907 = 710381447 -- 1x2 4x2 7x2 repeat
887 x 907 x 911 = 732907699 -- 7x2 9x3 repeat
907 x 911 x 919 = 759348563 -- 3x2 5x2 repeat
911 x 919 x 929 = 777767161 -- 1x2 6x2 7x5 repeat
919 x 929 x 937 = 799964687 -- 6x2 7x2 9x3 repeat
929 x 937 x 941 = 819115093 -- 1x3 9x2 repeat
937 x 941 x 947 = 834985999 -- 8x2 9x4 repeat
941 x 947 x 953 = 849244031 -- 4x3 repeat
947 x 953 x 967 = 872708797 -- 7x4 8x2 repeat
953 x 967 x 971 = 894826021 -- 2x2 8x2 repeat
967 x 971 x 977 = 917360989 -- 9x3 repeat
971 x 977 x 983 = 932539661 -- 3x2 6x2 9x2 repeat
977 x 983 x 991 = 951747481 -- 1x2 4x2 7x2 repeat
983 x 991 x 997 = 971230541 -- 1x2 repeat
991 x 997 x 1009 = 996919243 -- 9x4 repeat
2006-06-13 10:44:30 · answer #1 · answered by stellarfirefly 3 · 1⤊ 0⤋
there is none. to get a nine digit product, use the prime numbers 97,101,103,107,109,113,127, 131,137,139,149,151,157, 163,167,173,179,181, 191,193,197,199,211, and 223. No product of three consecutive primes in this range will reveal a 9 digit number without at least one repeating digit.
2006-06-13 09:11:45 · answer #2 · answered by dbs1226 3 · 0⤊ 0⤋
Thank you all for your answers. I, too, concluded that there wasn't a nine digit number that would fit the criteria. In desperation I tried base-16, as well. *sigh*
2006-06-15 03:13:05 · answer #3 · answered by Kalie H 1 · 0⤊ 0⤋
What do you mean by "whose digits are unique"? Do you mean that all of the digits are different?
2006-06-13 08:18:28 · answer #4 · answered by AnyMouse 3 · 0⤊ 0⤋