Given the fact that the Gilbreath's Conjecture has been a major topic of research in Aritmatic progression for well over a Century,and as bellow:2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 611 2 2 4 2 4 2 4 6 2 6 4 2 4 6 6 21 0 2 2 2 2 2 2 4 4 2 2 2 2 0 41 2 0 0 0 0 0 2 0 2 0 0 0 2 41 2 0 0 0 0 2 2 2 2 0 0 2 21 2 0 0 0 2 0 0 0 2 0 2 01 2 0 0 2 2 0 0 2 2 2 21 2 0 2 0 2 0 2 0 0 01 2 2 2 2 2 2 2 0 01 0 0 0 0 0 0 2 01 0 0 0 0 0 2 21 0 0 0 0 2 01 0 0 0 2 21 0 0 2 01 0 2 21 2 01 21The Gilbreath's conjecture in a way as easy and comprehensive as possible.He proposed that these differences, when calculated repetitively and left as bsolute values, would always result in a row of numbers beginning with 1,In this paper we bring elementary proof for this conjecture

Sazegar, Hashem

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ISSN 2347-1921                                                            3688 | P a g e                                                         J u n e  0 4 ,  2 0 1 5   AN  ELEMENTARY PROOF  OF GILBREATH’S CONJECTURE Hashem Sazegar Training department of National Iranian Gas company, Pipeline,Zone4 ABSTRACT Given the fact that the Gilbreath's Conjecture has been a major topic of research in Aritmatic progression for well over a Century,and as bellow: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 1 2 2 4 2 4 2 4 6 2 6 4 2 4 6 6 2 1 0 2 2 2 2 2 2 4 4 2 2 2 2 0 4 1 2 0 0 0 0 0 2 0 2 0 0 0 2 4 1 2 0 0 0 0 2 2 2 2 0 0 2 2 1 2 0 0 0 2 0 0 0 2 0 2 0 1 2 0 0 2 2 0 0 2 2 2 2 1 2 0 2 0 2 0 2 0 0 0 1 2 2 2 2 2 2 2 0 0 1 0 0 0 0 0 0 2 0 1 0 0 0 0 0 2 2 1 0 0 0 0 2 0 1 0 0 0 2 2 1 0 0 2 0 1 0 2 2 1 2 0 1 2 1  The Gilbreath’s conjecture  in a way as easy and comprehensive as possible. He proposed that these differences, when calculated repetitively and left as bsolute values, would always result in a row of numbers beginning with 1,In this paper we bring elementary proof for this conjecture. Keywords.Forward difference operator,finite difference method,difference Equation 2010 Mathematics Subject Classification:11A25,11R58         Council for Innovative Research Peer Review Research Publishing System Journal: JOURNAL OF ADVANCES IN MATHEMATICS Vol .10, No.7 www.cirjam.com , editorjam@gmail.com ISSN 2347-1921                                                            3689 | P a g e                                                         J u n e  0 4 ,  2 0 1 5   1  INTRODUCTION: Given the fact that the Gilbreath's Conjecture has been a major topic of research in aritmatic progression for well over a century,the Gilbreath conjecture in a way as easy and comprehensive as possible. Hopefully it will help the right person take this conjecture out of the unsolved list and into the list of accomplishments of mathematics. To begin the story, the anecdote goes that an undergraduate student named normanGilbreath was doodling on a napkin one day in a cafe and found a very interesting characteristic of the list of sequential prime numbers and the diferences between them. He proposed that these diferences, when calculated repetitively and left as absolute values, would always result in a row of numbers beginning with 1 (after the first row). No one has been able to prove it. In 1878, eighty years before Gilbreath's discovery, François Proth had, however, published the same observations along with an attempted proof, which was later shown to be false.Andrew Odlyzko verified that 𝑑1𝑘  is 1 for 𝑘 ≤ 𝑛 = 3.4 × 1011 in 1993, but the conjecture remains an open problem. Instead of evaluating n rows, Odlyzko evaluated 635 rows and established that the 635th row started with a 1 and continued with only 0's and 2's for the next n numbers. This implies that the next n rows begin with a 1,see[15] Notation We define 𝑑𝑛𝑘 is  K-th row, n- th,Number,  in 𝑑𝑛𝑘 =  𝑑𝑛+1𝑘−1 − 𝑑𝑛𝑘−1  We should prove that 𝑑1𝑘 = 1,for any k Theorem: 𝑑1𝑘 = 1,for any k  Proof: Assume that the Gilbreath's Conjecture  is correct until 𝑝𝑚 ,that is  m-th prime in first row by induction, we prove that this Conjecture is correct  for 𝑝𝑚+1,hence  below table is correct by induction 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61…….𝑝𝑚−2𝑝𝑚−1  𝑝𝑚  1 2 2 4 2 4 2 4 6 2 6 4 2 4 6 6 2 1 0 2 2 2 2 2 2 4 4 2 2 2 2 0 4 1 2 0 0 0 0 0 2 0 2 0 0 0 2 4 1 2 0 0 0 0 2 2 2 2 0 0 2 2 1 2 0 0 0 2 0 0 0 2 0 2 0 1 2 0 0 2 2 0 0 2 2 2 2 1 2 0 2 0 2 0 2 0 0 0 1 2 2 2 2 2 2 2 0 0 1 0 0 0 0 0 0 2 0 1 0 0 0 0 0 2 2 1 0 0 0 0 2 0 1 0 0 0 2 2 1 0 0 2 0 1 0 2 2 1 2 0 1 2 1 Notice that in above table for K- th row, n- th number,we have  𝑑𝑛𝑘 =  𝑑𝑛+1𝑘−1 − 𝑑𝑛𝑘−1 < 3𝑛 ≤ 3𝑚 ,Now we prove that this table is correct for 𝑝𝑚+1, 2  LEMMA: For  simplicity  this conjecture we state some Lemmas as below: Lemma 1: if  𝑝𝑚  to be m-th  prime ,so 𝑝𝑚 < 3𝑚  for 𝑚 ≥ 1 Proof: According to  1  ,this is  Correct  Lemma 2: the Second row is correct  ,i.e,  𝑝𝑚+1 − 𝑝𝑚  < 𝑝𝑚 < 3𝑚  ISSN 2347-1921                                                            3690 | P a g e                                                         J u n e  0 4 ,  2 0 1 5    Proof, this  is correct  by refer to [1] Lemma 3: the third row is correct ,i.e, 𝑑𝑚−13 =   𝑝𝑚+1 − 𝑝𝑚  −  𝑝𝑚 − 𝑝𝑚−1  < 𝑝𝑚−1 < 3𝑚−1, Proof :this  is correct by refer to [1] Lemma 4: k-th row is correct,4 ≤ 𝑘≤𝑚 + 1,i.e 𝑑𝑚− 𝑘−2 𝑘 =  𝑑𝑚− 𝑘−3 𝑘−1 − 𝑑𝑚− 𝑘−2 𝑘−1  < 3𝑚− 𝑘−2  Proof: we assume that this is not hold for 𝑘 ≥ 4,notice that  from 𝑘 = 4 to 𝑘 = 𝑚 + 1  , we have 𝑑𝑚− 𝑘−2 𝑘 =  𝑑𝑚− 𝑘−3 𝑘−1 −𝑑𝑚−𝑘−2𝑘−1≥3𝑚−𝑘−2 So for simplicity we write abbreviation as below: 𝑎1 = 𝑎 − 𝑏 ≥ 3𝑚−2 𝑎2 = 𝑎1 − 𝑏1 ≥ 3𝑚−3 . . . 𝑎𝑚−2 = 𝑎𝑚−3 − 𝑏𝑚−3 ≥ 3 𝑚−1 −(𝑚−2) We  add  above formula ,hence:  𝑎1 + 𝑎2 + ⋯ + 𝑎𝑚−2 ≥ 3 + 32 + ⋯ 3𝑚−2 But each item is smaller  than𝑎, and 𝑎 < 𝑝𝑚−1,So: Or                                            𝑝𝑚−1 >3𝑚−1−32(𝑚−2) According to  1 , there are constants numbers 𝑐1&𝑐2 such that: 𝑐1 𝑚 − 1 log 𝑚 − 1 < 𝑝𝑚−1 < 𝑐2 𝑚 − 1 log⁡(𝑚 − 1) So this is  Contradiction for 𝑚 ≥ 3 Lemma 5: suppose  the s-th  row  is correct 𝑠 ≥ 4 ,so , 𝑠 + 1 ≤ 𝑘 ≤ 𝑚 + 1, we  prove that 𝑑𝑚− 𝑘−2 𝑘 =  𝑑𝑚− 𝑘−3 𝑘−1 −𝑑𝑚−𝑘−2𝑘−1<3𝑚−𝑘−2 Proof :   this proof is similar to Lemma 4, by substitute  𝑚 − 𝑠 instead   𝑚 So ,                                          𝑝𝑚−𝑠−1 >3𝑚−𝑠−1−2𝑚−𝑠−2 This is  Contradiction for 𝑚 − 𝑠 ≥ 3 Corollary :By refer to above Lemmas,assume that  we have some equations  as below: 𝑝𝑘−2 ≤ 𝑎1 = 𝑎 − 𝑏 < 3𝑚−2 𝑝𝑘−3 ≤ 𝑎2 = 𝑎1 − 𝑏1 < 3𝑚−3 . . . 2 = 𝑝1 ≤ 𝑎𝑚−2 = 𝑎𝑚−3 − 𝑏𝑚−3 < 3 𝑚−1 − 𝑚−2  Then,𝑎𝑚−2 = 2,and this is contradiction ,because   𝑎𝑚−2,is odd Notice that ,if  from s-th row, we have  equations  like above , we reach to similar conclusion .  If   in  above  equations , g-th row to be changed,i.e , 3𝑘−𝑔 ≤ 𝑎𝑔−1 = 𝑎𝑔−2 − 𝑏𝑔−2 < 𝑝𝑘−𝑔 ,this is contradiction too. 3   MAIN THEOREM:  ISSN 2347-1921                                                            3691 | P a g e                                                         J u n e  0 4 ,  2 0 1 5   Theorem: 𝑑1𝑘 = 1,for any k Proof:According to above Lemmas this theorem is  hold, for 𝑘 = 𝑚 + 1 since 𝑑1𝑚 +1 < 3  then 𝑑1𝑚 +1 = 1,therefore  we proved this Theorem. References: [1] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford,1964. [2] Killgrove, R. B.; Ralston, K. E. "On a Conjecture Concerning the Primes." [3] "SWAC (Computer)."http://en.wikipedia.org/wiki/SWAC (computer) Accessed February15, 2012. [4] Weisstein, Eric W. "Proth's Theorem." From MathWorld{A Wolfram Web Resource. http://mathworld.wolfram.com/ProthsTheorem.html [5] Proth, Francois, Theoremes Sur Les Nombres Premiers, C. R. Acad. Sci. Paris, 1877, Volume 85, Pages 329-332, [6] Weisstein, Eric W. "Mertens Conjecture." From MathWorld{A Wolfram Web Resource. http://mathworld.wolfram.com/MertensConjecture.html [7] C. B. Haselgrove (1958). "A disproof of a conjecture of Polya." Mathematika, Volume 5 , Pages 141-145 doi:10.1112/S0025579300001480 [8] Caldwell, Chris K. "Largest Known Primes." Univery of Tennesse At Martin. http://primes.utm.edu/top20/page.php?id=3 [9] Caldwell, Chris K. "Probable Prime." Univery of TennesseAt Martin. http://primes.utm.edu/glossary/page.php?sort=PRP [10] Mulcahy, Colm. Card Colm. http://www.maa.org/columns/colm/cardcolm201008.html [11] Andrew Odlyzko: Home Page http://www.dtc.umn.edu/ odlyzko/ Accessed February 16, 2012. [12] Weisstein, Eric W. "Fermat's Little Theorem." From MathWorld{A Wolfram Web Resource. http://mathworld.wolfram.com/FermatsLittleTheorem.html [13] Weisstein, Eric W. "Mersenne Prime." From MathWorld{A Wolfram Web Resource. http://mathworld.wolfram.com/MersennePrime.html [14] C. B. Haselgrove (1958). A disproof of a conjecture of Polya. Mathematika, 5 ,pp 141-145 doi:10.1112/S0025579300001480 [15] Dubuque, Bill. (Number of oddly factored numbers <= n) - (number of evenly factored numbers <= n). July 03 2002. http://oeis.org/A072203. Accessed April 9, 2012. [16]An interesting Opportunety:theGilbreath Conjecture, www.carroll.edu/library/thesisArchive/Sturgill-Simon_2012final.pdf [17]Research Proposal, Elan A. Segarra, Harvey MuddCollege ,Senior ThesisMathematicsSpring(2005  

An Elementary Proof of Gilbreaths Conjecture

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An Elementary Proof of Gilbreaths Conjecture

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