THERE ARE INFINITELY MANY FIBONACCI COMPOSITES WITH PRIME SUBSCRIPTS

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1 Research and Communcatons n Mathematcs and Mathematcal Scences Vol 10, Issue 2, 2018, Pages ISSN Publshed Onlne on November 19, Jyot Academc Press THERE ARE INFINITELY MANY FIBONACCI COMPOSITES WITH PRIME SUBSCRIPTS Department of Mathematcs Nan Chang Unversty Nan Chang P R Chna e-mal: fenslu@126com Abstract From the entre set of natural numbers successvely deletng some resdue classes mod a prme, we nvent a recursve seve method Ths s a modulo algorthm on natural numbers and ther sets The recursvely sftng process mechancally yelds a sequence of sets, whch converges to the set of the certan subscrpts of Fbonacc compostes The correspondng cardnal sequence s strctly ncreasng Then the well known theory, set valued analyss, allows us to prove that the set of the certan subscrpts s an nfnte set, namely, the set of Fbonacc compostes wth prme subscrpts s nfnte 2010 Mathematcs Subject Classfcaton: Prmary 11N35; Secondary 11A07, 11Y16, 11B37, 11B50, 11A51, 11B39 Keywords and phrases: Fbonacc compostes, recursve seve method, order topology, lmt of sequence of sets Receved October 9, 2018

2 124 Fbonacc number 1 Introducton F x s defned by the recursve formula F 1 = F 2 = 1, 1, F x + 1 = Fx + Fx 1 From the aspect of prmalty, lke the Mersenne numbers, n the Fbonacc sequence there s a conjecture and an open problem There are nfntely many Fbonacc prmes There are nfntely many Fbonacc compostes wth prme subscrpts It s well known that the Fbonacc sequence s a dvsblty sequence, so we consder Fbonacc compostes wth prme subscrpts Today n analytc number theory, by the normal seve method, lke the twn prme conjecture, t s stll extremely dffcult f not hopeless to solve the above open problem But mathematcal research often off the beaten path In ths paper, we solve ths open problem by a recursve seve method In 1998, Drobot showed a theorem: f p > 7 s a prme such that p 2, 4 mod 5, and 2p 1 s also prme, then 2p 1 Fp [6], [9] For example: p = 19, 37, 79, 97, 139, 157, 199, 229, 307, 337, 367, 379, 439, 499, 547, 577, When p and 2 p + 1 both are prme, the prme p s sad to be a Sophe German prme

3 THERE ARE INFINITELY MANY FIBONACCI 125 When p and 2p 1 both are prme, the prme p s sad to be another Sophe German prme If we prove that there are nfntely many another Sophe German prmes of the form 5 k + 2, 5k + 4, then we prove that there are nfntely many Fbonacc compostes wth prme subscrpts F p In 2011, author used the recursve seve method, whch reveals some exotc structures for varous sets of prmes, to prove the Sophe German prme conjecture [3] Recently, author solved a smlar open problem: there are nfntely many Mersenne compostes wth prme exponents [5] Here we extend the above structural result to solve the open problem about Fbonacc compostes In order to be self-contaned we repeat some contents n the paper [3, 4, 5] 2 A Recursve Seve Method for Fbonacc Compostes For expressng a recursve seve method by well formed formulas, we extend both basc operatons addton and multplcaton +, nto fnte sets of natural numbers We use small letters a, x, t to denote natural numbers and captal letters A, X, T to denote sets of natural numbers except F For arbtrary both fnte sets of natural numbers A, B, we wrte A = a, a2,, a,, a, a1 < a2 < < a < < a 1 n n B = b, b2,, bj,, bm, b1 < b2 < < bj < < b We defne 1 m x, A B = a + b a + b,, a + b,, a + b, a + b + 1 1, 2 1 j n 1 m n m, AB = a b1, a2b1,, abj,, an 1b, a b 1 m n m

4 126 Example 7, 9 + 0, 10, 20 = 7, 9, 17, 19, 27, 29, 10 0, 1, 2 = 0, 10, 20 For the empty set 0/, we defne 0 / + B = 0/ and 0 / B = 0/ We wrte Let A \ B for the set dfference of A and B X A = a1, a2,, a,, a n mod a be several resdue classes mod a If gcd ( a, b) = 1, we defne the soluton of the system of congruences X A = a1, a2,, a,, an mod a, X B = b1, b2,, bj,, bm mod b, to be X D = d11, d21,, dj,, dn 1m, dnm mod ab, where x dj mod ab s the soluton of the system of congruences x a mod a, x b j mod b The soluton X D mod ab s computable and unque by the Chnese remander theorem For example, X D = 9, 17, 27, 29 mod 30 s the soluton of the system of congruences X 7, 9 mod 10, X 0, 2 mod 3

5 THERE ARE INFINITELY MANY FIBONACCI 127 The reader, who s famlar wth model theory, know that we found a model and formal system of the second order arthmetc [4] Here we do not dscuss the model and formal system From the entre set of natural numbers successvely deletng some resdue classes modulo a prme, and leave resdue classes, we nvented a recursve seve method or modulo algorthm on natural numbers and ther sets Now we ntroduce the recursve seve method for another Sophe German prmes of the form 5 k + 2, 5k + 4 Let p be -th prme, p 0 = 2 For every prme p, let B mod the soluton of the congruence Example x( x 1) 0 mod p 2 p be B 0 B 1 B 2 B 3 B 4 0 0, 2 0, 3 0, 4 0, 6 mod 2, mod 3, mod 5, mod 7, mod 11, B, ( p + 1) 2 mod p 0 Let m 0 = m 1 = m 2 = 5, 10, 30,

6 128 for all > 2, let m + 1 = pj 0 From the resdue class x 2, 4 mod 5 we successvely delete the resdue classes B 1 mod p 1,, B mod p, leave the resdue class T + 1 mod m +1 Then the left resdue class T + 1 mod m + 1 s the set of all numbers x of the form 5 k + 2, 5k + 4, such that x ( 2x 1) does not contan any prme p as a factor ( x ( 2x 1), 1 ) = 1 j p m + Let X D mod + 1 m be the soluton of the system of congruences X T mod m, X B mod p Let T + 1 be the set of least nonnegatve representatves of the left resdue class T + 1 mod m + 1 We obtan a recursve formula for the set T +1, whch descrbe the recursve seve method or modulo algorthm for another Sophe German prmes of the form 5 k + 2, 5k + 4 T 0 = 2, 4, T + 1 = ( T + m 0, 1, 2,, p 1 ) \ D (21) The number of elements of the set T + 1 s T + 1 = 2 ( pj 2) (22) 3 We exhbt the frst few terms of formula (21) and brefly prove that the algorthm s vald by mathematcal nducton

7 THERE ARE INFINITELY MANY FIBONACCI 129 The resdue class T 2, 4 mod 5 s the set of all numbers x of the 0 = form 5 k + 2, 5k + 4 Now the set X 2, 4 mod 5 s equvalent to the set X ( 2, , 1 = 2, 4, 7, 9 mod 10, from them we delete the soluton of the system of congruences D 1 = 2, 4 mod 10, and leave = ( 2, , 1 ) \ 2, 4 7, 9 T 1 = The resdue class T 1 mod 10 s the set of all numbers x of the form 5 k + 2, 5k + 4 such that ( x ( 2 x 1), 10) = 1 Now the set X 7, 9 mod 10 s equvalent to the set X ( 7, , 1, 2 ) = 7, 9, 17, 19, 27, 29 mod 30, from them we delete the soluton of the system of congruences D 2 = 9, 17, 27, 29 mod 30, and leave = ( 7, , 1, 2 ) \ 9, 17, 27, 29 7, 19 T 2 = The resdue class T mod 30 s the set of all numbers x of the form 5 k + 2, 5k + 4 such that ( x ( 2 x 1), 30) = 1 2 We delete nothng by the prme 5 from T 2 mod 30 So that let T 3 = T 2 The set T 3 mod 30 s equvalent to the set X 7, 19, 37, 49, 67, 79, 97, 109, 127, 139, 157, 169, 187, 199 mod 210 From them we delete D 3 7, 49, 67, 109 mod 210,

8 130 and leave T 4 19, 37, 79, 97, 139, 127, 157, 169, 187, 199 mod 210 The resdue class T mod 210 s the set of all numbers x of the form 4 5 k + 2, 5k + 4 such that ( x ( 2 x 1), 210) = 1 And so on Suppose that the resdue class T mod m, for > 2 s the set of all numbers x of the form 5 k + 2, 5k + 4 such that ( x ( 2 x 1), m ) = 1 We delete the resdue class B mod p from them In other words, we delete the soluton X D mod m + 1 of the system of congruences X T mod m, X Now the resdue class T mod B mod p m s equvalent to the resdue class ( T m, 1, 2,, p 1 ) mod m From them we delete the soluton D mod m + 1, whch s the set of all numbers x of the form 5 k + 2, 5k + 4 such that x( x 1) 0 mod p It 2 follows that the left resdue class T + 1 mod m + 1 s the set of all numbers x of the form 5 k + 2, 5k + 4, and ( x ( 2x 1), m + 1 ) = 1 Our algorthm s vald It s easy to compute T = 2T ( p 2) for > 2 by the above algorthm + 1 We may rgorously prove formulas (21) and (22) by mathematcal nducton, the proof s left to the reader In the next secton we refne formula (21) and solve the open problem

9 THERE ARE INFINITELY MANY FIBONACCI A Theorem About Fbonacc Compostes We call another Sophe German prmes p > 7 of the form 5 k + 2, 5k + 4 S-prmes Let T e be the set of all S-prmes T e = { x : x s a S-prme} We shall determne an exotc structure for the set lmt of a sequence of sets ( T ), lm T = Te, lmt = ℵ0 Then we prove that the cardnalty of the set theory of those structures, T e based on the T e s nfnte by well known T e = ℵ 0 Based on the recursve algorthm, formula (21), we successvely delete all numbers x of the form 5 k + 2, 5k + 4 such that x ( 2x 1) contans the least prme factor p We delete non S-prmes or non S-prmes together wth a S-prme The sftng condton or seve s x( 2x 1) 0 mod p p x For p > 7 we modfy the sftng condton to be x( 2x 1) 0 mod p p x (31) < Accordng to ths new sftng condton or seve, we successvely delete the set C of all numbers x, such that ether x or 2x 1 s composte wth the least prme factor p For p 7, C = { x : x X T mod m x( 2x 1) 0 mod p p x}

10 132 For p > 7, C = { x : x X T mod m x( 2x 1) 0 mod p p < x}, but reman the S-prme x f there s a x > 7 such that p = x n T mod m Note: If p = 3, 7, then 2 p 1 = 5, 13 are Fbonacc prmes F, both are deleted 5, F7 We delete all sets C j wth 0 j < from the set N 0 of all natural numbers x of the form 5 k + 2, 5k + 4, and leave the set The set of all S-prmes s L Te = = 1 N0 \ C j 0 N0 \ C 0 The recursve seve (31) s a perfect tool, wth ths tool we delete all non S-prmes and leave all S-prmes So that we only need to determne the number of all S-prmes T e If we do so successfully, then the party obstructon, a ghost n house of prmes, has been automatcally evaporated [8] Wth the recursve seve (31), each non S-prme s deleted exactly once, there s need nether the ncluson-excluson prncple nor the estmaton of error terms, whch cause all the dffculty n normal seve theory Let A be the set of all S-prmes x less than p A = { x : x < p x s S-prme}

11 THERE ARE INFINITELY MANY FIBONACCI 133 From the recursve formula (21), we know that the left set L s the unon of the set A of S-prmes and the resdue class T mod m, L = A T mod m (32) Now we ntercept the ntal segment T from the left set L, whch s the unon of the set A of S-prmes and the set T of least nonnegatve representatves Then we obtan a new recursve formula T = A T (33) Except remanng all S-prmes x less than T, both sets T and T are the same p n the ntal segment Formula (33) expresses the recursvely sftng process accordng to the sftng condton (31), and provdes a recursve defnton of the ntal segment T The ntal segment T s a well chosen notaton We shall consder some propertes of the ntal segment T, and reveal some structures of the sequence of the ntal segments ( T ) to determne the set of all S-prmes and ts cardnalty Let A be the number of S-prmes less than p Then the number of elements of the ntal segment T s T = A + T (34) From formula (22), we deduce that the cardnal sequence ( T ) s strctly ncreasng for all > 2 Based on order topology obvously we have T < T +1 (35) lmt = ℵ0 (36)

12 134 Intutvely we see that the ntal segment T approaches the set of all S-prmes T e, and the correspondng cardnalty T approaches nfnty as Thus the set of all S-prmes s lmt computable and s an nfnte set Next we gve a formal proof based on set valued analyss 31 A formal proof Let A be the subset of all S-prmes n the ntal segment T, A = { x T : x s S-prme} (37) We consder the structures of both sequences of sets ( T ) and ( A ) to solve the open problem Lemma 31 The sequence of the ntal segments ( T ) and the sequence of ts subsets ( A ) of S-prmes both converge to the set of all S-prmes T e Frst from set theory, next from order topology we prove ths lemma Proof For the convenence of the reader, we quote a defnton of the set theoretc lmt of a sequence of sets [2] Let ( F n ) be a sequence of sets, we defne lm supn= Fn and lm as follows: nfn= F n lm sup F n = Fn +, n= n= 0 = 0 lm nf n= F n = Fn + n= 0 = 0 It s easy to check that whch belongs to lm s the set of those elements x, supn= Fn F n for nfntely many n Analogously, x belongs to

13 THERE ARE INFINITELY MANY FIBONACCI 135 lm nfn= F n f and only f t belongs to n belongs to all but a fnte number of the F n If F for almost all n, that s t lm sup Fn n= = lm nf n= Fn, we say that the sequence of sets ( F n ) converges to the lmt lm Fn = lm sup Fn n= = lm nf n= Fn We know that the sequence of left sets ( L ) s descendng L 1 L2 L Accordng to the defnton of the set theoretc lmt of a sequence of sets, we obtan that the sequence of left sets ( L ) converges to the set T e lm L = L = Te (38) The sequence of subsets ( A ) of S-prmes s ascendng A 1 A2 A, we obtan that the sequence of subsets ( A ) converges to the set T e, lm A = A = Te (39) The ntal segment T s located between two sets A and L A T L Thus the sequence of the ntal segments ( T ) converges to the set T e lm T = Te (310)

14 136 Accordng to set theory, we have proved that both sequences of sets ( T ) and ( A ) converge to the set of all S-prmes T e lm T = lm A = Te (311) Next we prove that accordng to order topology both sequences of sets ( T ) and ( A ) converge to the set of all S-prmes T e We quote a defnton of the order topology [1] Let X be a set wth a lnear order relaton; assume X has more one element Let B be the collecton of all sets of the followng types: (1) All open ntervals ( a, b) n X (2) All ntervals of the form [ a 0, b), where a 0 s the smallest element (f any) n X (3) All ntervals of the form [ a, b0 ), where b 0 s the largest element (f any) n X The collecton B s a bases of a topology on X, whch s called the order topology Accordng to the defnton there s no order topology on the empty set or sets wth a sngle element The recursvely sftng process, formula (33), produces both sequences of sets together wth the set theoretc lmt pont T e X 1 X 2 : T 1, T2,, T, ; Te, : A 1, A2,, A, ; Te We further consder the structures of sets X 1 and X 2 usng the recursvely sftng process (33) as an order relaton < j T < Tj, ( T < Te ), < j A < A ( A < T ) j, e

15 THERE ARE INFINITELY MANY FIBONACCI 137 The set X 1 has no repeated term It s a well ordered set wth the order type ω + 1 usng the recursvely sftng process (33) as an order relaton Thus the set X 1 may be endowed an order topology The set X 2 may have some repeated terms We have computed out the frst few S-prmes x The set X 2 contans more than one element, may be endowed an order topology usng the recursvely sftng process (33) as an order relaton Obvously, for every neghbourhood ( c, T e ] of T e there s a natural number 0, for all > 0, we have T ( c, T e ] and A ( c, Te ], thus both sequences of sets ( T ) and ( A ) converge to the set of all S-prmes T e lm A = Te, lm T = Te Accordng to the order topology, we have agan proved that both sequences of sets ( T ) and ( A ) converge to the set of all S-prmes T e We also have lm T = lm A (312) The formula T s a recursve asymptotc formula for the set of all S-prmes T e In general, f T = 0/, the set X = { 0/ } only has a sngle element, e whch has no order topology In ths case formula (312) s not vald and our method of proof may be useless [3] Lemma 31 reveals an order topologcal structure and a set theoretc structure for the set of all S-prmes on the recursve sequences of sets By the well known theory of those structures, we easly prove that the cardnalty of the set of all S-prmes s nfnte 2

16 138 Theorem 32 The set of all S-prmes s an nfnte set We gve two proofs Proof We consder the cardnaltes T and A of sets on two sdes of the equalty (312), and the order topologcal lmts of cardnal sequences ( T ) and ( A ) as the sets T and A both tend to T e From general topology we know, f the lmts of both cardnal sequences ( T ) and ( A ) on two sdes of the equalty (312) exst, then both lmts are equal; f lm A does not exst, then the condton for the exstence of the lmt lm T s not suffcent [7] For S-prmes, the set T s nonempty T 0/, the formula (3,12) s e e vald, obvously the order topologcal lmts lm A and lm T on two sdes of the equalty (312) exst, thus both lmts are equal lm A = lm T From formula (36) lm T =, we have ℵ 0 lm A = ℵ0 (313) Usually, let π ( n) be the countng functon, the number of S-prmes 2 less than or equal to n Normal seve theory s unable to provde nontrval lower bounds of π 2( n) by the party obstructon [8] Let n be a natural number Then the number sequence ( m ) s a subsequence of the number sequence (n), we have the By formula (37), the We have lm π ( n) = lm π2( m ) 2 A s the set of all S-prmes less than m, and A s the number of all S-prmes less than m, thus π ( m ) = A lm π 2 ( m ) = lm A 2

17 THERE ARE INFINITELY MANY FIBONACCI 139 From formula (313), we prove lm π2( n ) = ℵ0 (314) We drectly prove that the number of all S-prmes s nfnte wth the countng functon Next we gve another proof by the contnuty of the cardnal functon Proof Let f : X Y be the cardnal functon f ( T ) = T from the order topologcal space X to the order topologcal space Y X Y : T 1, T2,, T, ; Te, : T1, T2,, T, ; ℵ0 It s easy to check that for every open set [ T 1, d ), ( c, d ), ( c, ℵ0 ] n Y the premage [ T, d), ( c, d), ( c, ] s also an open set n X So that 1 T e the cardnal functon T s contnuous at order topology T e wth respect to the above Both order topologcal spaces are frst countable, hence the cardnal functon T s sequentally contnuous By a usual topologcal theorem [1] (Theorem 213, p 130), the cardnal functon T preserves lmts lm T = lmt (315) Order topologcal spaces are Hausdorff spaces In Hausdorff spaces, the lmt pont of the sequence of sets ( T ) and the lmt pont of cardnal sequence ( T ) are unque We have proved Lemma 31, lm T = T, and formula (36), lmt = ℵ0 Substtute, we obtan that the set of all S-prmes s an nfnte set, e T e = ℵ 0 (316)

18 140 Wthout any estmaton or statstcal data, wthout the Remann hypothess, by the recursve algorthm, we well understand the recursve structure, set theoretc structure and order topologcal structure for the set of all S-prmes on sequences of sets We obtan a formal proof of the open problem n pure mathematcs By Drobot s theorem we have solved the open problem about Fbonacc compostes Theorem 33 There are nfntely many Fbonacc composte numbers wth prme subscrpts References [1] J R Munkres, Topology, (2nd Edton), Prentce Hall, Upper Saddle Rver (2000), 84 [2] K Kuratowsky and A Mostowsky, Set Theory: Wth an Introducton to Descrptve Set Theory, North-Holland, Publshng Company (1976), [3] Fengsu Lu, On the Sophe German prme conjecture, WSEAS Transactons on Mathematcs 10(12) (2011), [4] Fengsu Lu, Whch polynomals represent nfntely many prmes, Global Journal of Pure and Appled Mathematcs 14(1) (2018), [5] Fengsu Lu, There are nfntely many Mersnne composte numbers wth prme exponents, Advances n Pure Mathematcs 8(7) (2018), DOI: [6] L Somer, Generalzaton of a theorem of Drobot, Fbonacc Quarterly 40(5) (2000), [7] Mchel Hazewnkel, Encyclopeda of Mathematcs, Lmt [8] Terence Tao, Open Queston: The Party Problem n Seve Theory [9] V Drobot, On prmes n the Fbonacc sequence, Fbonacc Quarterly 38(1) (2000), g