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Sequences of Primes Obtained by the Method of Concatenation

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SEQUENCES OF PRIMES OBTAINED BY THE METHOD OF CONCATENATION (COLLECTED PAPERS) Copyright 2016 by Marius Coman Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 Peer-Reviewers: Dr. A. A. Salama, Faculty of Science, Port Said University, Egypt. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Mohamed Eisa, Dept. of Computer Science, Port Said Univ., Egypt. EAN: 9781599734668 ISBN: 978-1-59973-466-8 1 INTRODUCTION The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (as, for instance, the prime terms in Smarandache concatenated odd sequence, defined as the sequence obtained through the concatenation of the first n odd numbers, or the terms obtained concatenating two primes of the form 6k + 1 or of the form 6k – 1 in Smarandache prime- partial-digital sequence, defined as the sequence of prime numbers which admit a deconcatenation into a set of primes), but also on “Smarandache-Coman sequences of primes”, defined by the author as “all sequences of primes obtained from the terms of Smarandache sequences using any arithmetical operation”; the SC sequences presented in this book are related, of course, to concatenation, but in three different ways: the S sequence is obtained by the method of concatenation but the operation applied on its terms is some other arithmetical operation (e.g. the case of SC sequence of primes obtained through a recurrence relation between the terms of Smarandache consecutive numbers sequence, which itself is defined through concatenation), the S sequence is not obtained by the method of concatenation but the operation applied on its terms is concatenation (e.g. the case of the SC sequence of primes obtained concatenating with digit 1 the terms of Smarandache proper divisor products sequence), or both S sequence and SC sequence are using the method of concatenation (e.g. the case of the SC sequence of primes obtained concatenating with the digit 1 the terms of Smarandache reverse sequence, defined as the sequence obtained through the concatenation of the first n positive integers in reverse order). Part Two of this book, “Sequences of primes obtained by the method of concatenation”, brings together 51 articles which aim, using the mentioned method, to highlight sequences of numbers that are rich in primes or are liable to lead to large primes. The method of concatenation is applied to different classes of numbers, e.g. squares of primes, Poulet numbers, triangular numbers, reversible primes, twin primes, repdigits, factorials, primorials, in order to obtain sequences, possible infinite, of primes. Beside other particular conjectures stated in these papers, a generally applicable conjecture states that for any arithmetic progression a + b*k, where at least one of a and b is different than 1, that also satisfies the conditions imposed by the Dirichlet’s Theorem (a and b are positive coprime integers) is true that the sequence obtained by the consecutive concatenation of the terms a + b*k has an infinity of prime terms. If this conjecture were true, the fact that the Smarandache consecutive numbers sequence 1, 12, 123, 1234 (...) could have not any prime term (thus far there is no prime number known in this sequence, though there have been checked the first about 40000 terms) would be even more amazing. Part Two of this book also contains a paper which lists a number of 33 sequences of primes obtained by the method of concatenation, sequences presented and analyzed in more detail in my previous papers, gathered together in five books of collected papers: “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, “Two hundred and thirteen conjectures on primes”, “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function”, “Sequences of integers, conjectures and new arithmetical tools”, “Formulas and polynomials which generate primes and Fermat pseudoprimes”. 2 SUMMARY Part One. Primes in Smarandache concatenated sequences and Smarandache- Coman sequences 1. Sets of primes distinguished among the terms of twenty Smarandache concatenated sequences 2. Four conjectures on the Smarandache prime partial digital sequence 3. Poulet numbers in Smarandache prime partial digital sequence and a possible infinite set of primes 4. Conjecture on an infinity of subsequences of primes in Smarandache prime partial digital sequence 5. Two conjectures on Smarandache’s proper divisor products sequence 6. Conjecture on the primes S(n) + S(n + 1) – 1 where S(n) is a term in Smarandache- Wellin sequence 7. Conjecture on the primes S(n + 1) + S(n) – 1 where S(n) is a term in the concatenated odd sequence 8. Four Smarandache type sequences obtained concatenating numbers of the form 6k – 1 respectively 6k + 1 9. The definition of Smarandache reconcatenated sequences and six such sequences 10. On terms of consecutive numbers sequence concatenated both to the left and to the right with same prime 11. Primes obtained concatenating with 1 to the left the terms of three Smarandache sequences 12. Primes obtained concatenating to the right with 1 the terms of concatenated n-th powers sequences Part Two. Sequences of primes obtained by the method of concatenation 1. A list of 33 sequences of primes obtained by the method of concatenation 2. Sixteen sequences of primes obtained by concatenation from p – 1 respectively p + 1 where p prime 3. Four conjectures on the numbers 2*(p*q*r) ± 1 where p and q = p + 6 and r = q + 6 are odd numbers 4. Two conjectures involving the numbers obtained concatenating repeatedly odd multiples of 3 with 111 5. Conjecture involving the numbers obtained concatenating the square of a prime p with p then with 1 6. Two conjectures involving the numbers obtained concatenating a prime p with 9 then with p itself 7. Conjecture on the numbers obtained concatenating two primes p and q where q – p + 1 also prime 8. Four conjectures on the numbers p, 2*p – 1, 3*p – 10 and n*p – n + 1 where p prime 9. Three conjecture on the numbers obtained concatenating p^2 with (p^2 + 1)/2, p + 12, p^2 + 12 10. Three conjecture on the primes obtained concatenating p with (p – 1)/2 respectively with (p + 1)/2 where p prime 3 11. Primes obtained concatenating a prime p to the left with 3 and to the right with a square of prime q^2 12. Primes obtained concatenating a square of prime to the left with 24*k + 4 and to the right with 3 13. Conjecture on the infinity of primes obtained concatenating a prime p with p + 30*k 14. Conjecture on a set of primes obtained by a formula involving reversible primes and concatenation 15. Primes obtained concatenating even numbers n with 0 then with n + 2 then again with 0 then with n + 5 16. Primes of the form (6k – 1)]c[(6k + 1)]c[(6 k – 1) and (6k + 1)]c[(6k – 1)]c[(6k + 1) where “]c[“ means “concatenated to” 17. Primes obtained concatenating p – 1 with q^2 where p and q are primes or Poulet numbers 18. Primes obtained concatenating two primes with the same digital root respectively digital sum 19. Primes obtained concatenating the prime factors of composite numbers 20. On the numbers (n + 1)*p – n*q where p and q primes, p having the group of its last digits equal to q 21. Four conjectures on the primes p^2 + 18*m and q^2 – 18*n between the squares p^2, q^2 of a pair of twin primes 22. Three sequences obtained concatenating P – 1 with 1 and 11 respectively P + 1 with 11 where P Poulet numbers 23. Two conjectures on the quintets of numbers (p, p + 10, p + 30, p + 40, p + 60) 24. Two conjectures on the quintets of numbers (p, p + 20, p + 30, p + 50, p + 80) 25. Three conjectures on the primes obtained concatenating 30*k with 30*k + p where p prime 26. An unusual conjecture on primes involving concatenation and repunits 27. Primes obtained concatenating p repeatedly with 6 then with q where (p, q) are sexy primes 28. Primes of the form p]c[x]c[q]c[y]c[r where p, q, r consecutive primes, q – p = x and r – q = y 29. Conjecture on the pairs of primes (p, q = p + k) involving concatenation 30. Conjecture on the consecutive concatenation of the numbers n*k + 1 where k multiple of 3 31. Primes obtained concatenating p prime with p + 2 and p + 6 respectively with p + 4 and p + 6 32. Four conjectures on the triplets [p, p + 2, p + 8] and [p, p + 6, p + 8] where p prime 33. Four conjectures on the triplets [p, p + 4, p + 10] and [p, p + 6, p + 10] where p prime 34.
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