An infinite primality conjecture for prime-subscripted Fibonacci numbers

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1 Notes on Number Theory and Discrete Mathematics ISSN Vol. 2, 205, No., 5 55 An infinite rimality conjecture for rime-subscrited Fibonacci numbers J. V. Leyendekkers and A. G. Shannon 2,3 Faculty of Science, The University of Sydney NSW 2006, Australia 2 Faculty of Engineering & IT, University of Technology Sydney, NSW 2007, Australia 3 Camion College PO Box 3052, Toongabbie East, NSW 246, Australia s: t.shannon@warrane.unsw.edu.au, Anthony.Shannon@uts.edu.au Abstract: The row structures of the rime-subscrited Fibonacci numbers in the modular ring Z 4 show distinction between rimes and comosites. The class structure of the Fibonacci numbers suggest that these row structures must survive to infinity and hence that Fibonacci rimes must too. The functions F = K ± and F (factors) = k ± suort the structural evidence. The grah of (K/k) versus dislays a Raman-sectra form ersisting to infinity: ln(k/k) is linear in in the comosite case while rimes lie along the -axis to infinity. Keywords: Fibonacci numbers, Prime numbers, Comosite numbers, Modular rings, Raman sectra. AMS Classification: B39, B50. Introduction On knowing the infinite, Franklin has this to say: It is evident that the idea of an infinite structure cannot be derived urely from ercetual exerience [...] Our ercetual exerience is finite in character [2]. This aer will exlore an asect of infinity in the context of Fibonacci rimes and the regularity of the Fibonacci numbers generated from the second order homogeneous recurrence relation F, n = Fn + Fn 2 n > 2, (.) which gives rise to the many eriodicities found in this sequence [4] and its very recise integer structure; for examle, in the associated modular ring Z 4 (Table ). 5

2 Row r i Class i 4 Comments N = 4r i + i even, 2 4 N N, n 2n ( ) odd,3 4 4 ; Table. Classes and rows for Z 4 2n N 4 This articular ring is the most aroriate in this context because the rime-subscrited Fibonacci numbers satisfy [, 5] 2 2 F = F + + F (.2) 2 and the only odd class in Z 4, which can form this sum of squares is 4, which is generated by 4r + [2]; that is, rime-subscrited Fibonacci numbers will always fall in this class. When F is itself rime, Equation (.2) is the only ossible sum [2], but not with comosites [7, 8]. 2 2 Class structures with Fibonacci numbers The rime-subscrited Fibonacci numbers will now be considered in detail in order to assess the evidence of infinitely many Fibonacci rimes. Since there are infinitely many rimes, F will have infinitely many values, but since F may be comosite, the number of rimesubscrited Fibonacci rimes may be finite. The class structure for the Fibonacci numbers in the modular ring Z 4 is (2.) which is reeated to infinity as the formation of a recursive sequence does not change [5]. Since F 4, the row structure is given by F = 4r +. (2.2) The row structures of F for = 7 to 0 are dislayed in Table 2 according to *, the rightend-digit of (that is, (mod 0)). This * defines the class of in the modular ring Z 5 [8, 2, 8]. As can be seen there is a distinction between rimes and comosites which is worth exloring further. If there are no rimes for very large, then the row structure of F would be very restricted for * = or 3, and some row structures for * = 7 or 9 would not occur. In view of the recise nature of the F structure this would not be ossible. This makes a comelling case for infinitely many rime Fibonacci numbers. 52

3 * Primes Comosites 4 4, 44, 4, , 44, 4 7 4, Table 2. Row structures of F, 444, 4, 44, 4, 4 3 Functions and factors Here we consider functions and factors [0, ] defined resectively by F = K ± and F (factors) = k ±. When K/k for 7 to 0 is lotted as function of, the result is a Ramanlike sectra [3] with a base of unity (since K = k for rimes), and a variety of bands when K/k (reresenting the comosites). Continuous bands without the base of would have to occur if no more rimes occur for large. This would be inconsistent with any normal sectra and would indicate severe ruture of the F -sequence structure which would not be ossible in view of the formation mechanism of the Fibonacci recurrence relation. ln(k/k) as a function of : for comosite F this function is linear and asses through the origin where K = k and ln(k/k) = 0; that is, for comosites only ln( K / k) (3.) 0( ϕ ) in which ϕ is the Golden Ratio [4]. In Table 3 the standardized values reresented by ln(k/k) and /0(ϕ-), resectively are set out to show that the two sides of (3.) are roortionally aroximate. The following standard normalization formula was utilized [6] x x x i min i = (3.2) xmax xmin ln(k/k) /0(ϕ-) Table 3. Comosite F 53

4 When F is rime the line coincides with the -axis which meets the comosite line at the origin and increases asymtotically. Another arameter which suorts this structural evidence is S [3]: S = F i i= = F +2 (3.3) from which, S *, the right-end-digits for S, for rimes are distinct from those of comosite F (Table 4). * Primes S * Comosites 3 6, 8 3 5, 9 4 7, 4, Table 4. S, 3 97 The stability of S * is based on the structural ability of the Fibonacci numbers from their very definition by means of a linear recurrence relation (.) and the eriodicity of their class structure in the modular ring (2.). 4 Concluding comments This class attern of F is invariant since the mechanism of generation of the Fibonacci numbers remains the same to infinity. If class structure is invariant, then the demonstrated difference between the row structures for rimes and comosites should also be invariant. The relationshis of K and k with also show that rimes are generated as long as rimes exist; that is, since there is an infinity of rimes [9], then there must also be an infinity of rimesubscrited Fibonacci rime numbers. Another ossible line of aroach for further research related to the central issue in this aer would be by means of asymtotic roofs; that is, for almost all n. These have been used reviously for Fibonacci numbers by Horadam and Subba Rao [4, 6, 7]. References [] Drobot, V. (2000) On Primes in the Fibonacci sequence. The Fibonacci Quarterly. 38(), [2] Franklin, J. (204) An Aristotelian Realist Philosohy of Mathematics: Mathematics as the Science of Quantity and Structure. New York, Palgrave Macmillan,. 40. [3] Hoggatt, V. E. Jr. (969) Fibonacci Numbers. Boston, MA, Houghton Mifflin. 54

5 [4] Horadam, A. F. (966) Generalization of Two Theorems of K. Subba Rao. Bulletin of the Calcutta Mathematical Society. 58, [5] Kno, K. (990) Theory and Alication of Infinite Series. New York, Dover. [6] Larsen, R. J., & Marx, M. L. (200) An Introduction to Mathematical Statistics and Its Alications. Uer Saddle River, NJ: Prentice Hall, [7] Leyendekkers, J. V., & Shannon, A. G. (998) Fibonacci Numbers within Modular Rings. Notes on Number Theory & Discrete Mathematics. 4(4), [8] Leyendekkers, J. V., & Shannon, A. G. (203) The Structure of the Fibonacci Num-bers in the Modular Ring Z 5. Notes on Number Theory & Discrete Mathematics. 9(), [9] Leyendekkers, J. V., & Shannon, A. G. (203) The Pascal-Fibonacci Numbers. Notes on Number Theory & Discrete Mathematics. 9(3), [0] Leyendekkers, J. V., & Shannon, A. G. (204) Fibonacci Primes. Notes on Number Theory & Discrete Mathematics. 20(2), 6 9. [] Leyendekkers, J. V., & Shannon, A. G. (204) Fibonacci Number Sums as Prime Indicators. Notes on Number Theory & Discrete Mathematics. 20(4), [2] Leyendekkers, J. V., Shannon, A. G. & Rybak, J. M. (2007) Pattern Recognition: Modular Rings and Integer Structure. North Sydney, Raffles KvB Monograh No.9. [3] Lilley, T. H. (973) Raman Sectroscoy of Aqueous Electrolyte Solution. In Felix Franks (ed.) Water: A Comrehensive Treatise. Volume Three. New York: Plenum Press, Ch. 6. [4] Livio, M. (2002) The Golden Ratio. New York, Broadway Books. [5] Somer, L. (2002) Generalization of a Theorem of Drobot. The Fibonacci Quarterly. 40(4), [6] Subba Rao, K. (954) Some Proerties of Fibonacci Numbers. I. Bulletin of the Calcutta Mathematical Society. 46, [7] Subba Rao, K. (959) Some Proerties of Fibonacci Numbers II. Mathematics Student. 27, [8] Terr, D. (996) On the Sums of Digits of Fibonacci Numbers. The Fibonacci Quarterly. (4), [9] Watkins, J. J. (204) Number Theory: A Historical Aroach. Oxford: Princeton University Press, , 50,