GPF SEQUENCES Lisa Scheckelhoff CAPSTONE PROJECT

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1 MATHEMATICS BONUS FILES for faculty ad studets GPF SEQUENCES Lisa Schecelhoff CAPSTONE PROJECT Sprig 2006 Departmet of Mathematics OHIO NORTHERN UNIVERSITY Advisor Dr. Mihai Caragiu

2 ( ) gpf SUMMARY: Let be the greatest prime factor of the iteger. We will use the fuctio to geerate a class of sequeces of primes depedig o the parameters p (a prime) ad ab, (positive itegers) i the followig way: () = p + = gpf ( a + b), =,2,3,... We will eplore several properties of these "liear gpf sequeces" ad we will see that we eter a territory of etremely iterestig cojectures, of a similar flavor with the celebrated Collatz problem. I some cases we fid otrivial coectios with the Mersee primes. Several MATLAB-geerated plots illustratig our cojectures will be provided, as well as a complete proof for the special case ad b a = arbitrary (this special case is i a surprisig coectio with a recet Clay Prize-awarded result of Be Gree ad Terrece Tao).

3 . INTRODUCTION: A MOTIVATING PROBLEM Collatz s cojecture For ay iteger a, defie the sequece { } N i the followig way: N N + = a 3N +, if Nis odd = N, if Nis eve 2 COLLATZ CONJECTURE: For every a, the resultig sequece { N } N evetually eters the cycle Paul Erdős o the Collatz cojecture: "Mathematics is ot yet ready for such problems." (Erdős offered $500 for whoever solves it) 2

4 Figure : A plot of the Collatz sequece startig a = 27 with. Notice that there is a substatial icrease util the sequece evetually settles ito the cycle,4,2,,... I what follows we will try to itroduce classes of sequeces of a similar flavor with the sequeces i the Collatz problem. They will be based o the greatest prime factor fuctio ad have the same limit-cycle property. 3

5 2. LINEAR GPF SEQUENCES NOTATION: gpf ( ) = the greatest prime factor of. ab, For every two itegers ad every prime we defie a sequece of primes { i the followig way: } p = p + = gpf ( a + b), Let us call such sequeces, Liear gpf sequeces I what follows we will ivestigate the behavior of the lieargpf sequeces i which ab, = p term. are fied, while various choices are made for the first Let us deote the class of these sequeces by GPF( ab, ). 4

6 A VERY SIMPLE EXAMPLE: If we pic = 3for a sequece i GPF(,), the first few terms of the sequece will be 3,7,2,3,2,3,... Note that the sequece is ultimately periodic. Figure 2: Plot of the first 0 terms of the sequece i GPF(,) with = 3. 5

7 THIS APPEARS TO BE A SPECIAL CASE OF A VERY GENERAL PATTERN: THE GPF CONJECTURE: All liear ultimately periodic. gpf sequeces are 8, with = ,3,83,9,7,37,097,3,049,09,97,37 Figure 3: The sequece { } GPF( ) with period, is ultimately periodic 6

8 Figure 4: Logarithmic plot of a sequece { } GPF ( ) It is ultimately periodic with period with. 4,3 = 9 54,7577,2467,2657,3720,49, ,4,577,808,499,29,5809, 9623,37,444,64,9,2677,03, ,7,997,607,850,700,9807,05557,93 7

9 We do t have to choose very large cotaiig large umbers! ab, to get gpf sequeces with periods As a eample we ca tae the sequece { } GPF( ) 6,5 with = 2. Figure 5: Logarithmic plot of a sequece { } GPF ( ) 6,5 with. = 2 The first few terms are: 2, 7, 07, 647, 23, 3, 83, 503, 3023, 843, 08863, 9749, 37, 827, 4967, 727, 397, 3, 9, 5, 69, 367, 2207, 09, 2, 4, 25, 5, 93, 63, 6983, 4903, 244, 23, 3, 83, 503, 3023, 843, 08863, 9749, 37 (the period is i bold italics). 8

10 3. MULTIPLE LIMIT CYCLES OBSERVATION: I some cases we may have more tha oe limit cycle ab, (for give we may get two distict periods which do ot differ by a simple circular permutatio). GPF Figure 6: ( 2,6) PERIOD: 5,, = 29 3,7 Figure 7: GPF 2,6, = 73 ( ) PERIOD: 3,,9 9

11 4. GPF DIGRAPHS Defie directed graphs ( digraphs ) G ab, G ab as follows: The vertices of are the prime umbers, There is a arrow (directed edge) from pto gpf ( ap + b) FACT: The out-degree of every verte p i G ab, is. 5. G, AND MERSENNE PRIMES G, : EDGES: p gpf ( p + ) No matter with which prime we start, the flow of the 2 3 evetually lead us ito the cycle. G, edges will NOTE: 2 epoet p is a edge i G, if ad oly if p = 2 (with a which is implicitly a prime), that is, if ad oly if pis a 2 G, Mersee prime. Thus, the hypothesis The i degree of i is ifiite is equivalet to the log-stadig cojecture There are ifiitely may Mersee primes. A GENERALIZED MERSENNE CONJECTURE G p, The i degree of i is ifiite for every prime p 0

12 The gpf cojecture i terms of GPF digraphs If we follow the flow of directed edges, startig from ay verte i a GPF digraph we evetually get ito a limit cycle. G 6. - THE SOPHIE GERMAIN DIGRAPH 2, VERTICES { 2,3,5,7,,3,... }- ALL PRIMES EDGES p gpf ( 2p + ) G 2, LIMIT CYCLE: Startig from ay iitial prime, if we follow the flow defied by the directed edges of the Sophie Germai digraph, we will eter, after a sufficietly large umber of iteratios, i the termial cycle idicated above.

13 Figure 8: Distace to the Sophie Germai limit cycle startig from primes up to,000,000 Figure 9: Logarithmic plot for the sequece GPF 2, with ( ) = 779. Amog the primes up to a millio, 779 is the oe which requires the largest umber of iteratios i order to get ito the Sophie Germai cycle. 2

14 G,2 7. THE TWIN DIGRAPH EDGES: p gpf ( p + 2) 2 2 FACTS. There is a loop (which i itself is a isolated coected compoet) ad a uique cycle (3 cycle) The edge p gpf ( p 2) ( p, p+ 2) is a twi pair. 3. If (, 2) 3 + is icreasig if ad oly if p p+ is a twi pair, the with the oly eceptio of p = 3, the edge emergig from p + 2 is decreasig. 4. Startig from ay odd prime p, the edge flow of,2 will lead us ito the (uique) 3 cycle G 3 3

15 8. LOOPS IN G ab, As a special case, we ca ivestigate the loops i the digraphs G ab, p p For a prime p, the eistece of the loop p the eistece of a iteger with gpf ( ) that ap + b = p. That is, ( ) with ( ) b = p a gpf p G ab, p i is equivalet to p such By usig the above relatio it follows that, give a prime pthere are G p p ifiitely may digraphs ab, i which the loop appears. 2 For eample, the loop appears i all digraphs G ab, with 22 ( r ) 2 G 5,6 b= a (oe eample would be ). 4

16 9. PROVING THE GPF CONJECTURE IN THE SPECIAL CASE a = I our paper [3] we maaged to prove the followig result: The cojecture is true for primes{ } THEOREM GPF a =. That is, ALL sequeces of satisfyig + = gpf ( + b) are ultimately periodic. PROOF: Let pbe the least prime such that p b. That is, p p tha. bad bis at the same time divisible by all the primes smaller LetM bp = +. 5

17 MAIN LEMMA: Let { } ( b) that. The there eists a itegerl such that. M GPF, ad assume > l < PROOF OF THE MAIN LEMMA: The umbers + + ( ), b,..., p b + ub, + vb 0 u< v p, ay two of them, say with caot be all primes. Ideed, ote that caot give the same remaider whe divided by p. Otherwise their differece ( vb) ( ub) ( v u) p sice p bit would follow that p v u, a cotradictio, sice0 < v u < p. Sice they are pumbers givig differet remaiders whe divided by p larger tha p, sice M = bp+ > p) must be divisible by p ad thus is composite. Let j {,..., p } + + = b will be divisible by, ad it follows that oe of them (which must be be miimal with the property that + jb composite. It follows that the umbers ( ), it follows that, +,..., + ( ), + b,..., + j bare prime, ad by usig the fact that { } GPF(, b) are terms of the sequece{ } : b,..., ( j ) + + j is b j b = + = + b Let us estimate the term gpf ( b) gpf ( jb) = + = +. + j + j 6

18 Note that the composite iteger + jbcaot have prime divisors that are less tha p, sice bis divisible by all primes smaller tha p, ad o the other had > p. Therefore ( ) = gpf + jb + j + p jb At the same time, the iequality followig equivalece chai holds: + p jb < is true. Ideed, the + jb b p < + jb < p jb < ( p ) < p j By the choice of ( pb+ >b), the left had side of the last iequality is strictly less tha, while the right had side is at least. This cocludes the proof of the Mai Lemma, if we tael = + j END OF THE PROOF OF THE THEOREM: Let A { prime, M} = < be the set of primes less tha M. By repeatedly applyig the Mai Lemma we coclude that if there eists a iteger such that A. l > l M the 7

19 Therefore the attractor set A will be visited ifiitely may times by the. recurret sequece { } Sice A IS FINITE we ca fid l l 2 implies that { } is ultimately periodic. < with l = l. This automatically 2 0. A CONNECTION WITH THE GREEN-TAO THEOREM The 2004 Clay Research Award wet to Bejami Gree, who, i a joit wor with Terrece Tao, solved a logstadig cojecture, to the effect that for ay 3there are ifiitely may term progressios of primes [5]. Their result is a major breathrough i our uderstadig of primes. Applyig the Gree-Tao result to our GPF sequeces we get, as a cosequece, the followig result: 3 b GPF, b such that < 2 <... < For every there eists a iteger ad a GPF sequece { } ( ) For eample, if = 23, we ca select b = = The the first 23 terms i the sequece GPF, are precisely < = + b< = + 2 b<... < = + 22 b { } ( ) (result due to Marus Frid, Paul Joblig ad Paul Uderwood []) ad 8

20 . THE REDUCTION THEOREM If the GPF cojecture is true for prime sequeces satisfyig = + ( + ) gpf ( ad bd ) gpf a b the it is also true for prime sequeces = + d 2 + satisfyig, where is a fied iteger. PROOF OF THE REDUCTION THEOREM: Let K = gpf ( d ). We have ( + ) = ma, ( + ) ( ) gpf ad bd K gpf a b We wat to show that assumig all prime sequeces satisfyig ( ) gpf a b = + + are ultimately periodic, the all prime sequeces satisfyig are ultimately periodic too. ( ( )) y = + ma K, gpf ay + b Ideed, if the terms of a prime sequece { } y satisfyig ( ( )) y = + ma K, gpf ay + b 9

21 y > K { } 0 y 0 satisfies the recurrece relatio y gpf ( ay b) satisfy for all, the the sequece ultimately periodic. Otherwise for some s agai the ultimate periodicity of { } + actually = + ad therefore is < t we will have ys = yt = K, which implies y. This cocludes the proof of the Reductio Theorem. 2. COROLLARIES COROLLARY : If all prime sequeces satisfyig gpf a b ab,,gcd ab, ( )where ultimately periodic, the the GPF cojecture is true. + = + ( ) =are COROLLARY 2: The GPF cojecture is true for all prime sequeces satisfyig = gpf ( a + + ab), where ab,. 20

22 REFERENCES [] 23 primes i arithmetic progressio, [2] Bugeaud, Y., O the greatest prime factor of (ab+)(bc+)(ca+), Acta Arith. 86 (998), o., [3] Mihai Caragiu ad Lisa Schecelhoff, The greatest prime factor ad related sequeces, JP Joural of Algebra Number Theory ad Applicatios - to appear. [4] Corvaja, P. ad Zaier, U., O the greatest prime factor of (ab+)(ac+),proc. Amer. Math. Soc. 3 (2003), o. 6, [5] Be Gree, Terece Tao, The primes cotai arbitrarily log arithmetic progressios, [6] Hooley, C., O the greatest prime factor of a quadratic polyomial, Acta Math. 7 (967), [7] Hooley, C., O the greatest prime factor of a cubic polyomial, J. Reie Agew. Math. 303/304 (978), [8] Liu, H. Q., The greatest prime factor of the itegers i a iterval, Acta Arith. 65 (993), o. 4, [9] Le, M., A ote o the greatest prime factors of Fermat umbers, Southeast Asia Bull. Math. 22 (998), o.,