WON plate116 | World!OfNumbers [ October 9, 2001 ] An ODD_DDO puzzle question1 3 5 7 9 This WONplate is totally dedicated to integers composed of only ODD digits and whose prime factors are iDem DitO ! Is this a worthy challenge for you ? Sequence A062016 will give you a good starting pointfor the semiprime case (with exactly two prime factors). Bear in mind that more than two prime factors are allowed. Note that I am looking for ever larger integers/prime factors that are sufficiently random. I don't want to highlight the obvious ones that could be found too easily by reverse engineering like for instance 3 * Repunit(n). or that could be derived from an infinite pattern like (9)n5 = 5 * 1(9)n Lastly if the digitlength of the factors are > 1 and near equal to each other it certainly will enhance the beauty of the all-odd number found ! Solutions Two prime factorsInteger ?all_odd = f1 * f2(possible if f1 and f2 > 7 ?) Three prime factors 7335737 = 191 * 193 * 19915995731573 = 1153 * 1933 * 7177 Four prime factors111333 = 3 * 17 * 37 * 59 Five prime factors7977553773 = 3 * 7 * 17 * 113 * 197753 Etc. [ June 5, 2002 ]Jean Claude Rosa (email) solved the small question for 2 factors. The answer is no. Every prime factor larger than 7 and 'all odd' always ends in one of the following 20 numbers : 11, 13, 17, 19, 31, 33, 37, 39, 51, 53, 57, 59, 71, 73, 77, 79, 91, 93, 97 or 99. If we multiply two by two any pair of these 20 numbers we get the result that The digit at the tens position is always even. So for the first case with two prime factors there exist no solution for factors larger than 7 of course. Thus the products of two prime factors 'all odd' ends in 01, 03, 07, 09, 21, 23, 27, 29, 41, 43, 47, 49, 61, 63, 67, 69, 81, 83, 87 or 89. If we next multiply two by two any pair of these 20 numbers we still get the same result The digit at the tens position is always even. This means there is also no solution for the caseswith 4, 6, 8, ... prime factors 'all odd' and larger than 7. Additional Question from Jean Claude Rosa [ June 10, 2002 ]" Is it possible to find a palindromic number 'all odd' as a resultof the product of 3, 5, ... prime factors 'all odd' ? " [ July 15, 2002 ] Jean Claude Rosa calculated all the products of 3, 5 and 7 prime factors 'all odd' taken amongst 10 consecutive 'all odd' prime numbers from 3 up to 3,999,999,979. Here are the largest 'all odd' composites that he found With 3 prime factors 391133599733953317333733 = 73131517 * 73131599 * 73133351 With 5 prime factors 775793373119 = 191 * 193 * 197 * 317 * 337 With 7 prime factors 3353535515 = 5 * 11 * 17 * 31 * 37 * 53 * 59 He also found some cute little 'all odd' palindromes ! 595 = 5 * 7 * 17 3553 = 11 * 17 * 19 19591 = 11 * 13 * 137 " J'espère avoir un peu de temps pour continuer à rechercher des palindromes plus grands... " [ July 16, 2002 ] This afternoon I (J. C. Rosa) discovered larger 'all odd' palindromes, which of course made me very happy... Here they are With 3 prime factors 9979531359799 = 19 * 71171 * 7379951 With 4 prime factors 9935335335399 = 3 * 191 * 5171 * 3353153 With 5 prime factors 9997159517999 = 19 * 53 * 59 * 937 * 179579 With 6 prime factors 9919959599199 = 3 * 13 * 17 * 59 * 317 * 799991 He also found various palindromic composites with prime factors NOT all different 19399999391 = 59 * 59 * 5573111 19975357991 = 53 * 53 * 7111199 And finally this curious one ! 777555777 = 3 * 37 * 73 * 95959 [ July 21, 2002 ] J. C. Rosa communicates that the two largest 15-digit 'all odd' palindromes consisting of 3 or 5 distinct prime factors are 993957131759399 = 977 * 13537 * 75153751 995393999393599 = 13 * 19 * 53 * 193 * 393971573 [ October 21, 2002 ] Edwin Clark (email) (site) looked for 'all odd' palindromeswith 'all odd' & 'palindromic' prime factors using Maple. 9 = 3 * 3 33 = 3 * 11 393 = 3 * 131 939 = 3 * 313 33933 = 3 * 11311 39993 = 3 * 13331 55 = 5 * 11 77 = 7 * 11 99 = 3 * 3 * 11 1331 = 11 * 11 * 11 3993 = 3 * 11 * 11 * 11 3773 = 7 * 7 * 7 * 11 17377371 = 3 * 3 * 11 * 191 * 919 5775 = 3 * 5 * 5 * 7 * 11 Clearly some of these generalize. For example when 1333...3331 is a prime we get one of the form 3999...9993. The largest prime of this form I found was one with 95 digits. [ For this I used Maple's 'isprime' function to determine primality.] Unfortunately all of these have at least one small prime factor. I found these by putting in a list the 43 odd palindromic primes less than 10^5 and found all products of 2, 3, 4 and 5 of these. Then I checked the products for oddness and palindromicity. p.s. primes of the form 1333...3331 are PDP primes Plateau and Depression Primes and are since then more extensively researched. The second largest known probable prime of the form (12*10^n-21)/9 or 4*(10^n-1)/3-1 is 1(3)179551 thus creating an all odd palindrome of 17957 digits ! A000116 Prime Curios! Prime Puzzle Wikipedia 116 Le nombre 116
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