The sum of squares for primes

Transcription

1 Notes on Number Theory and Discrete Mathematics ISSN Vol., 05, No. 4, 7 The sum of squares for rimes J. V. Leyendekkers and A. G. Shannon,3 Faculty of Science, The University of Sydney, NSW 006, Australia Faculty of Engineering & IT, University of Technology, Sydney, NSW 007, Australia 3 Camion College, PO Box 305, Toongabbie East, NSW 46, Australia s: t.shannon@ camion.edu.au, Anthony.Shannon@uts.edu.au Abstract: Only rime integers that are in Class4 of the Modular Ring Z 4 equate to a sum of squares of integers x and y. A simle equation to redict these integers is develoed which distinguishes rime and comosite numbers in that one (x, y) coule exists for rimes, but comosites have either one coule with a common factor or the same number of coules as there are factors. In articular, comosite Fibonacci numbers always have multile (x, y) coules because the factors are all elements of 4. Keywords: Modular rings, Fibonacci sequence, Prime numbers, Comosite numbers, Rightend-digits, Pascal-Fibonacci numbers. AMS Classification: B39, B50. Introduction Fermat showed that odd integers which equal a sum of squares have the form 4r +, which is obvious in the modular ring Z 4 [6] (Table ). If we consider N = x + y with x odd and y even, then we see that odd squares are elements of class 4 and even squares are elements of 04 only. Row r i Class i Comments N = 4r i + i even 04, ( n n, ) Table. Classes and rows for Z 4 N N odd 4, 34 ; n N 4 7

2 We then have the two ossibilities = 4, N 4 N canbe a sumof squares, , N 34 N cannot be a sumof squares; that is, only rime numbers which are elements of 4 can be a sum of squares. In this note we develo a means of calculating x and y raidly in the sense of exerimental mathematics []. Calculations Thus for 4, If A = x + y, then which gives an uer limit of (.) = x + y = ( x + y) xy. (.) A ± A x, y = (.) for A. x (odd) and y (even) are the two solutions of (.). We now show that ( A ) * {, 5, 9}. The rimes,, can be divided into four tyes according, *, to their right-end-digits (REDs) [cf. 8]; that is, according to their classes in the modular ring Z 5 [, 6]. * {, 3, 7, 9} corresonding to classes5,35, 5, 45 so that the class of A* in Z 5 will be determined by *, but odd squares only have REDs equal to,5,9 (Table ). * A* A *, 9 3, 5, 9, 5 7 3, 5, 7 5, 9 9 3, 7 9 Table. Restraints on REDs 3 Examles of x, y in 4 In the sirit of Phillis [9], samles of rimes with calculated A, x and y are dislayed in Tables 3, 4, 5, 6. A x y A x y Table 3. * =, A* =, 9 8

3 A x y A x y Table 4. * = 3, A* =, 5, 9 A x y A x y Table 5. * = 7, A* = 3, 5, 7 A x y A x y Table 6. * = 9, A* = 3, 7 4 Fibonacci rimes The x, y values for Fibonacci rimes, F, are obtained simly from [7] F = F + F (4.) + in which F =, 5 (4.) the Binet equation [7], or from the Pascal-Fibonacci equation [3, 4] 9

4 i F = + (4.3) i= i from which the individual Pascal-Fibonacci (PF) numbers, N n, can be given by i. N = (4.4) i For examle, for = 7, F = 597, and the Binet and Pascal-Fibonacci equations (4.) and (4.3) then yield resectively F = From (.) and (4.) we get + = 597, = 597. A = F + F. (4.5) For examle, if =, then F = 89 = F + F = That is, A = 3 so that ( ) 5 = x = and y = (3 + 3) = 8, as required (Table 6). 5 Comosites as sums of squares As with rime integers, only odd comosite integers in class4 can equal a sum of squares [6], but the number of (x, y) ordered airs will be the same as the number of rime factors. N Factors Classes A x y Table 7. * = 7; A* = 3, 5, 7 If the factors all come from class 34, then there will be no sum of squares, but if some factors come from class 4, then there can be a sum of squares [6]. In some cases there is only

5 one ordered air but, unlike the rimes, x and y have a common factor in this case. Some examles are dislayed in Table 7 for * = 7. Similar results may be found for * =, 3 or 9. Note that 57 ( 47), 537 (3 79) and 597 (3 99) all have factors in 34 only, so there are no sums of squares. For comosite Fibonacci numbers with rime subscrits there are as many ordered airs (x, y) as there are factors. For examle, F 3 =, 346, 69 has factors, 557 4, 47 4, and (x, y) ordered airs: (987, 60) and (875, 76); F 37 = 4, 57, 87 has 3 factors, all in class 4, and thus 3 (x,y) ordered airs: (48, 584), (4909, 4) and (3859, 3044). For all the F values identified as comosite (F 9 to F 97 ) the factors are all in class 4 [5] so that there are multile (x,y) ordered airs for these integers [3]. 6 Comosites as sums of squares These distinct differences between rime and comosite numbers in class 4 is useful for rimality testing in that class, articularly as the integers increase in value the number of stes needed to find A do not necessarily increase. For instance, for * = 9 (Table 6), = 7589 takes only one ste while = 7669 takes six stes, whereas for = 53469, only four stes are need to obtain A. References [] Borwein, J, & D. Bailey. (004) Mathematics by Exeriment: Plausible Reasoning in the st Century. Natick, MA: A K Peters. [] Leyendekkers, J. V. & A. G. Shannon (0) The Modular Ring Z 5. Notes on Number Theory and Discrete Mathematics. 8(), [3] Leyendekkers, J. V. & A. G. Shannon (03) Fibonacci and Lucas Primes. Notes on Number Theory and Discrete Mathematics. 9(), [4] Leyendekkers, J. V. & A. G. Shannon (03) The Pascal-Fibonacci Numbers. Notes on Number Theory and Discrete Mathematics. 9(3), 5. [5] Leyendekkers, J. V. & A. G. Shannon (04) Fibonacci Primes. Notes on Number Theory and Discrete Mathematics. 0(), 6 9. [6] Leyendekkers, J. V., A. G. Shannon, & J. M. Rybak (007) Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograh No.. [7] Livio, M. (00) The Golden Ratio. New York: Golden Books. [8] Omey, E., S. Van Gulck. (05) What are the last digits of...? International Journal of Mathematical Education in Science and Technology. 46(), [9] Phillis, G. M. (005) Mathematics is not a Sectator Sort. New York: Sringer.