Random Sequences Using the Divisor Pairs Function

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1 Radom Sequeces Usig the Divisor Pairs Fuctio Subhash Kak Abstract. This paper ivestigates the radomess properties of a fuctio of the divisor pairs of a atural umber. This fuctio, the atecedets of which go to very aciet times, has radomess properties that ca fid applicatios i scramblig, key distributio, ad other problems of cryptography. It is show that the fuctio is aperiodic ad it has excellet autocorrelatio properties. Keywords. Divisor pairs fuctio, Radomess INTRODUCTION Oe of the earliest well-articulated mathematical problems to be foud i the world literature is that of divisors of a umber []. The problem of divisor pairs of a umber is metioed i the secod milleium BCE Saskrit text called the Śatapatha Brāhmaṇa. This scietific kowledge described i the text icludes geometry, algebra, ad astroomy ad it is oe of the earliest sources for Pythagorea triples []. The idea of divisors i the text is i the sese that 4 has three divisors (,,4) ad two divisor pairs (,4) ad (,). Speakig of divisors of a umber, the text states that the umber 7 has 5 divisor pairs ad 8 has 3 a a a3 divisor pairs. These umbers are correct sice the umber of divisors, d (), for = p p p3, equals ( a + )( a + )( a3 + ), ad the umber of divisor pairs which we call δ (), will be half of it but rouded up whe d() is odd. I geeral, δ () = d () / For the first example from the Śatapatha Brāhmaṇa, 7 = 4 3 5, so d (7) = 5 3 = 3 ad δ (7) = 5. Similarly, δ (8) = 3. Parethetically, it should be metioed that these umbers are metioed for their sigificace as the umber of days i the half-moth ad the moth [3]. a a a3 a The umber δ () is related to v (), the valecy of the umber = p p p3... p, that is defied to be v ) = a + a + a a. Figure gives the value of the fuctio δ () for <. ( 3 The fuctio v() satisfies the relatio: v (ab) = v (a) + v (b). Below is a list of begiig atural umbers with valecy of,, 3, ad so o: v () =:,, 3, 5, 7,, 3, (prime umbers) v () =: 4, 6, 9,, 4, 5, (squares ad semi-primes) v () =3: 8,, 8,, 7, v () =4: 6, 4, 36, 4, 54, Fuctios of v() are well kow i the mathematics literature [4],[5]. Related also to the valece umber is the umber of primes factors b (), which is simply equal to. Some scholars who do ot cosider the iteral astroomical evidece i the text date it to the first half of first milleium BCE.

2 δ ( ) Aother related fuctio is κ( ) = ( ) for the sequece of umbers. We also defie S( ) = κ ( i), which is the ruig sum of the κ() fuctio. Table provides the values of v (), d (), δ (), κ (), ad S () for Figure. The fuctio δ () for <, Table. Values of v (), d (), δ (), κ (), ad S () v () d () δ () 3 3 κ () S () I geeral, we ca cosider other fuctios of the expoets associated with the prime factors of a umber. We ca thus speak of a geeralized valece fuctio, V (), which is give by: V ( ) = f ( a, a,..., a) Figure. The fuctio S () for < 5

3 Thus δ () is a special case of V (). I geeral, it would be worthwhile to determie which forms of V () are of most iterest to the computer scietist from the poit of view of geeratig radom sequeces. δ ( ) Here we wish to study the radomess properties of the biary sequeceκ ( ) = ( ). This is cotiuatio of a project to examie the radomess characteristics of a variety of umber-theoretic fuctios which iclude prime reciprocals [6]-[8], Pythagorea triples [9], permutatio trasformatios [], ad Goldbach sequeces []. We show that the sequece is irratioal ad it ca be used as a pseudoradom sequece. A RELATED FUNCTION The Liouville fuctio λ() is a biary fuctio of v () that maps eve values to ad odd values to -: λ () = ( ) v( ) Table presets a compariso of the values of κ () ad λ () for 6. Table. Compariso of κ () ad λ () κ () λ () As see i Table, κ() ad λ () are differet at values of = 8, 6. Likewise, the values will be differet for = 7, 8, ad so o. The Liouville fuctio λ() satisfies the followig property: λ( d) = d, = perfect square, otherwise A umber with high valecy is composite i multiple ways. It is iterestig that Ramauja worked o highly composite umbers [] ad this work has attracted recet attetio [3]. A iteger is said largely composite if for m, d(m) d(). Ramauja [] preseted the followig result relevat to. If σ (N) deotes the sum of the iverses of the sth powers of the divisors of, the / { ( p p p3... pn) } σ ( N) < ( p )( p )...( p For s=, σ ( ) = d(n) is the umber of divisors of N. N ) 3

4 THE FUNCTIONS δ () AND κ () The properties of the divisor pairs fuctio δ () are obviously derivable from that of the divisor fuctio d ( ). These two fuctios satisfy the followig properties: d ( ) = d( ) d( ) if gcd(, ) = d( ) d( ) d( ) if gcd(, ) δ ( ) < δ ( ) δ ( ) if gcd(, ) δ ( ) δ ( ) δ ( ) if gcd(, ) = δ ( ) = δ ( ) δ ( ) if gcd(, ) = ad both d ( ) ad d ( ) are eve. For the fuctio κ () the followig properties are evidet (where p is prime): κ ( p) = κ ( p ) =, κ ( p ) =, κ ( p p) = =,4,5,8,9,... =,3,6,7,,,... Figure 3 presets the values of the fuctio κ () for < Figure 3. The fuctio κ () for < Theorem. There is o periodicity associated with the κ () fuctio for ay > N. Proof. We establish this result by assumig it is true ad the showig that leads to a cotradictio. Let the κ() sequece have a period of k after =N. This would imply that κ ( N ) = κ( N + k) ad κ( N + r) = κ( N + r + k) for ay r. Choose N+r = p so that ( p + k) = p p. This coditio that k = p p p for a radom p, which is that a umber may be writte as a semiprime mius a prime, is true from experimetal 4

5 calculatios ad also for large umbers [4]. This would imply that κ p) = κ( p ) κ( p ) or κ ( p) = κ( ), which is impossible. ( This theorem establishes that κ() is a irratioal fuctio ad, therefore, it ca be used as a pseudoradom sequece [5]. The autocorrelatio fuctio captures the correlatio of data with itself. For a data sequece a() of N poits the autocorrelatio fuctio C(k) is represeted by N C( k) = a( j) a( j + k) N j= For a oise sequece, the autocorrelatio fuctio C(k) = E(a(i)a(i+k)) is two-valued, with value of for k= ad a value approachig zero for k for a zero-mea radom variable. Sice S() drifts towards icreasig positive values, for ay choice of N, it would have a o-zero mea μ associated with it. Assumig ergodicity, such a sequece will have C(k) as for k= ad approximately μ for o-zero k. Figures 4 ad 5 preset the autocorrelatio fuctio of the series κ () for = ad 5, respectively. The value of μ =.36 ad μ 5 =.46. The value of C(k) for o-zero k is therefore cetered aroud.6 ad.3, respectively. p Figure 4. The autocorrelatio fuctio for κ () for =

6 Figure 5. The autocorrelatio fuctio for κ () for =5 As N becomes large the variace of the values i the autocorrelatio fuctio will reduce ad i the limit it will be zero. CONCLUSIONS This paper examied the properties of the divisor pairs fuctio. I particular, the biary sequece δ ( ) κ( ) = ( ), which is closely related to the Liouville fuctio, was ivestigated for its radomess characteristics. While its ruig sum drifts to positive values, its autocorrelatio fuctio is approximately two-valued which meas that it ca fid applicatios i may cryptography applicatios. May iterestig questios remai: These iclude behavior of S() for large values of ad the use of other fuctios of the valecy of a umber to geerate radom sequeces. Ackowledgemet. This research was supported i part by research grat #768 from the Natioal Sciece Foudatio. REFERENCES [] S. Kak, Early record of divisibility ad primality. [] A. Seideberg, The origi of mathematics. Archive for History of Exact Scieces.8: 3-34, 978. [3] S. Kak, The astroomy of Vedic altars. Vistas i Astroomy 36: 7-4, 993. [4] J. Cassaige, S. Fereczi, C. Mauduit, J. Rivat, A. Sàrközy, O fiite pseudoradom biary sequeces III: the Liouville fuctio, I. Acta Arith. 87: , 999. [5] C. Mauduit, Fiite ad ifiite pseudoradom biary words. Theoretical Computer Sciece 73: 49 6,. [6] S. Kak ad A. Chatterjee, O decimal sequeces. IEEE Trasactios o Iformatio Theory IT-7: , 98. [7] S. Kak, Ecryptio ad error-correctio codig usig D sequeces. IEEE Trasactios o Computers C-34: 83-89, 985. [8] S. Kak, New results o d-sequeces. Electroics Letters 3: 67, 987. [9] S. Kak, Pythagorea triples ad cryptographic codig. arxiv:4.377 [] S. Kak, O the mesh array for matrix multiplicatio.. arxiv:.54 [] K.R. Kachu ad S. Kak, Goldbach circles ad balloos ad their cross correlatio. arxiv:9.46 [] S. Ramauja, Highly Composite Numbers. Proc. Lodo Math. Soc. Series 4: 347 4, 95. [3] J.-L. Nicolas ad G. Robi, Highly composite umbers of S. Ramauja, The Ramauja Joural : 9-53, 997. [4] J.R. Che, O the Represetatio of a Large Eve Iteger as the Sum of a Prime ad the Product of at Most Two Primes. II. Sci. Siica 6: 4-43, 978. [5] G.H. Hardy ad E.M. Wright, A itroductio to the theory of umbers. Oxford: Claredo Press,

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