Prime Numbers in Generalized Pascal Triangles

Transcription

1 Prime Numbers in Generalized Pascal Triangles G. Farkas, G. Kallós Eötvös Loránd University, H-1117, Budapest, Pázmány Péter sétány 1/C, Széchenyi István University, H-9026, Győr, Egyetem tér 1, Abstract: In this paper we generalize the Pascal triangle and examine the connections among the generalized triangles and powering integers respectively polynomials. We emphasize connections with the binomial and multinomial theorems, and we present some computational results in topics of prime numbers and prime factorization. Some prime numbers (candidates) connected to the generalized Pascal triangle are presented. We describe a very efficient primality proving algorithm called ECPP (Elliptic Curve Primality Proving) and for one of the candidates an exact primality proof is given. Since this prime number has more than 1000 digits, it is a so-called titanic prime. Keywords: Pascal triangle, Generalized binomial coefficient, Primality testing, ECPP 1. Generalized Pascal Triangles 1.1 Introduction The interesting and really romantic Pascal triangle has a number of generalizations. We can construct the generalized Pascal triangles of s th order (or kind s), from the generalized binomial coefficients of order s. This idea was first published in 1956 by J. E. Freund [4]. The so-called Pascal pyramid is constructed from trinomial coefficients, which occur in the expansions (x + y + z) n (first mentioned by E. B. Rosenthal, 1960, [1]). Each of the outer faces of the pyramid are Pascal triangles. We can extend this idea to the multidimensional case, so Pascal hyperpyramids can be constructed from multinomial coefficients [1]. 109

2 Our generalization 1 is based on the well-known fact (see e.g. the paper of R. L. Morton from 1964, [9]) that from the n th row of the Pascal triangle with positional addition we get the n th power of 11 (Figure 1), where n is a non-negative integer, and the indices in the rows and columns run from =11 0, 11=11 1, 121=11 2, 1331=11 3, 14641=11 4, =11 5, Figure 1: Powers of 11 in the Pascal triangle This comes directly from the binomial equality n 10 0 n n 10 1 n 1 n 10 2 n 2 n n 1 n n 1 0 n (1) 1.2. Generalization Let us construct triangles, in which the powers of other numbers appear. To achieve this, let us consider the Pascal triangle as the 11-based triangle, and take the following definition. Definition Let 0 a 0, a 1,, a m-1 9 be integers. We get the k th element in the n th row of the a 0 a 1 a m-1 -based triangle if we multiply the k-m th element in the n-1 th row by a m-1, the k-m+1 th element in the n-1 th row by a m-2,, the k th element in the n-1 th row by a 0, and add the products. If for some index i we have k m + i < 0 or k m + i > n(m - 1) (i.e., an element in the n-1 th row does not exist) then we consider this element to be 0. The indices in the rows and columns run from 0 (Figure 2) Figure 2: The 257-based triangle 1 The main idea was presented by the second author in 1993, and later in a subsequent paper in 1997 ([6], [7]). Although the construction of triangles of coefficients in expansions of (a + bx) n and (a + bx + cx 2 ) n have already been mentioned by few other authors in the last years (see e.g. [10]), systematic analysis was published first only in [6] and [7]. 110

3 Remark Triangles with base-number 11 1 (s pieces of 1 digits) have already been known for decades as generalized Pascal triangles of s th order [1]. However, they were introduced using a combinatorial approach, by the generalization of the binomial coefficients [4]. The generalized binomial coefficient n (2) m s is the number of different ways of distributing m objects among n cells where each cell may contain at most s-1 objects [1]. This is the element in the m th column of the n th row in the generalized Pascal triangles of s th order. For s = 2 we get the "normal" binomial coefficients and the Pascal triangle. Now we summarize the main results about the general triangles. The proofs are here omitted. Details can be found in [7]. Theorem From the n th row of the a 0 a 1 a m-1 -based triangle by positional addition we get the n th power of the number a 0 a 1 a m-1. Example In the second row of the 257-based triangle = = Proposition The elements in the n th row of the general triangle are exactly the coefficients of the polynomial (a 0 + a 1 x + + a m-1 x m-1 ) n, where the k th element is the coefficient of x k. Example From the third row of the 257-based triangle (Figure 2) (2 + 5x + 7x 2 ) 3 = x + 234x x x x x 6. These results show that we have the "right" to call the new triangles as generalized Pascal triangles, since their general properties are very similar to those of the Pascal triangle. With deeper analysis we are able to discover a nice connection of the general triangle with the multinomial (sometimes referred as the polynomial) theorem. Definition For the digits of the base-number number a 0 a 1 a m-1 let the weight of a digit be its distance from the centerline. So w(a 0 ) = -w(a m-1 ), w(a 1 ) = -w(a m-2 ),. If the base number is odd, then w(a (m-1)/2 ) = 0. Let the unit of the weights be the distance of two neighboring elements in the triangle, i.e., w(a i ) = w(a i+1 ) + 1. As the elements of the triangle are sums, consider the parts of them. For such an expression let the weight of the part be the sum of the weights of its digits. If a digit is on the i th power, then we count its weight i-times. Theorem The elements in the n th row of the a 0 a 1 a m-1 -based triangle are exactly such sums of the coefficients of the polynomial (a 0 x 0 + a 1 x a m-1 x m-1 ) n, in which the weights of the parts are identical. 111

4 1.3. Investigations We can investigate a lot of interesting properties of general triangles. E.g. we can generalize further properties of the Pascal triangle, we can analyze divisibility results and modulo n coloring and calculate fractal dimensions. Some of these topics are partly covered in [6] and in [7]. Here we present a summary about the topics of prime numbers and factorizations in general triangles Prime numbers It is a trivial result that in triangles with base 11a 2 a 3 we get all of the prime numbers in the first position of the rows (similarly, as in the Pascal triangle). For general triangles it follows from the multinomial theorem and our Theorem that we can find prime numbers in given "directions" (details in [6]). E.g. in the abc-based triangles the middle position is interesting. Investigations on computer show that small primes are relatively common; however large primes are very rare. We can find all of the small primes up to 101 in the first 3 rows of the abc-based triangles in positions (2, 2) and (3, 3), except for 47, which occur first in the third row of the 1231-based triangle. The most common small prime is 13, which occur in triangles 112, 116, 132, 211, 213, 231, 312 and 611 in positions (2, 2) and (3, 3), by solving equations ac b and 13 6abc b, for a, b and c. R. C. Bollinger proved for triangles with base 11 1 (generalized Pascal triangle of order p) that for large n, "almost all" coefficients in the n th row are divisible by p (details in [1]). It is a conjecture that we have a similar result for our general triangles (e.g. in the 112-based triangle "almost all" coefficients are divisible by 5 and 7, in the 113- based triangle "almost all" coefficients are divisible by 7 and 11). Thus, we have only very little chance to find a large prime number in an arbitrary triangle. With a computer program e.g. in triangles with base numbers we can find 4 primes (candidates) more than 500 decimal digits checking the first 1100 rows (Tabular 1). Base Position Digits , , , , Table 1: Primes (candidates) in triangles with base numbers Factorizations Analyzing factorizations of numbers in general triangles on a computer, we find usually a few small factors some of which are repetitive, and a lot of composite parts, which are only possible to be decomposed with a large effort. Pure large prime factors are very rare (in 1 percent of the elements, or even less). 112

5 E.g. in the 112-based triangle we have found 7 pure large prime factors (candidates) more than 600 digits in the middle positions up to the 1750 th row (Tabular 2). Position Prime factor digits 1048, , , , , , , Table 2: Pure large prime factors (candidates) in the 112-based triangle The factorization of element (1726, 1726) is as follows: (5)*(7)^2*( ) Here the last number is the 1002-digit prime candidate. Our goal was to prove the prime property of this candidate. The importance of this proof was to overcome the 1000 decimal-digit limit, since after S. Yates we call these primes titanic ones. He started to collect the list of such numbers and their founders in the mid 80s. 2. Primality Testing 2.1. Introduction As a matter of fact, "primality test" is a procedure for deciding whether a positive integer n is prime or not. Finding an efficient algorithm for solving the above mentioned problem has been occupying the attention of lots of famous mathematicians for hundreds of years. 113

6 We have to distinguish two kinds of primality test methods. An exact primality test gives one of the following two answers: "n is a prime" or "n is a composite number". If we use a probabilistic primality test, the first answer is "n may be a prime". The running time of the probabilistic test is shorter, but we could not be hundred percent sure of the prime property of n. Although the exact tests are exclusively suited for primality proving, the probabilistic tests are very useful for producing so-called "candidates (for primality)". The following statement published by Euler, known as Fermat's Little Theorem, was the basic idea of the probabilistic primality tests: Theorem Let p be a prime and a an arbitrary integer. Then a p a (mod p). In particular, if p does not divide a then a p 1 a (mod p). Unfortunately, there are such n composite integers for that the congruence a n 1 a (mod n) is valid. In general if n is an odd composite number which is relatively prime to a and satisfies the above congruence, then n is a pseudoprime for the base a. We speak about a Carmichael number, if n is a pseudoprime for all bases to which they are relatively prime. Luckily, Carmichael numbers are less than up to Elliptic Curve Primality Proving (ECPP) Due to the total lack of space we give just a schematic description of the oldest and one of the newest exact primality proving methods (see the number-theoretical details in [2] and additional information about the algorithms and their implementation in [3] and [5]). The following routine, which checks the divisors of n from 2 to n, was the first algorithm for primality testing: 1 for d 2 to n 2 do if d n 3 then return COMPOSITE 4 return PRIME Let us observe that with this algorithm it takes a very long time to carry out the primality test for large numbers n. These days the most efficient exact primality testing algorithms are based on the theory of elliptic curves. The basic idea rises from the following theorem which is well-known in Group Theory: Theorem Let n be a candidate for primality and assume that we have a group (mod n), denoted by G. Let G d be the restricted group (mod d) and e the identity in G. If we can find an x G and an integer m satisfying the following conditions, then n is a prime: 114

7 1. m > G q for any prime q, where q n and q < n, 2. x m = e, 3. For each prime p dividing m, some coordinate of x (m/p) e is relatively prime to n. Note that for an integer n and a positive integer m the expression (m/p) is the Jacobi symbol which is the generalization of Legendre symbol to composite numbers. The appropriate group can be constructed with elliptic curves. Let K be one of the fields Q, R, C and Fq, where q is a prime number and Fq is the appropriate finite field. Let us consider the polynomial x 3 + ax + b K[x] from which we can get the base set of the group in the following way: Definition Let E(K) be the set of those points (x, y) K 2 which satisfy the equation y 2 = x 3 + ax + b. Then we say that E(K) is an elliptic curve over K. Note that if 4a b 0 then x 3 + ax + b has three distinct roots. Naturally, we need an identity element in E(K), therefore, let us complete it with the point infinitely far north denoted by O. To get a group we need to define a binary operation on E(K) which is associative and for each element a E(K) there exists an inverse element -a such that a + (-a) = (-a) + a = O. To do this, we can make use of such an extraordinary property of the elliptic curves that if a non-vertical line intersects it at two points, then it will also have a third point of intersection. It can be proved that this third point can be calculated in a similar way. Using additive notation we define the + operation by (x 1, y 1 ) + (x 2, y 2 ) = (x 3, -y 3 ), if (x 2, y 2 ) (x 1, -y 1 ). Thus, the sum of two points is not the third intersection point, but the reflection across the x-axis of it which is still on the same elliptic curve. Let us observe that if (x 2, y 2 ) = (x 1, -y 1 ), these two points define a vertical line which has no third intersection point, thus in this case (x, y) + (x, -y) = (x, -y) + (x, y) = O. It can be proved further that if (x 1, y 1 ), (x 2, y 2 ) are rational points than so is (x 3, -y 3 ). Now, we can define the appropriate group: Definition For a given elliptic curve y 2 = x 3 + ax + b, where 4a b 0, let E denote the group of rational points on the curve together with the identity O with the binary operation +. We present the assertion on which the elliptic curve primality tests are based. Theorem Let n N, (6, n) =1, E n is an elliptic curve over Z/nZ and m, s such integers for that s m. Let us assume that there exists such a point P E n for which m m P O and P O q 115

8 are valid for every prime factor q of s. Then for every prime divisor p of n the congruence E n 0 (mod s) is true, and if (3) then n is a prime number. s 4 n 1 2, Here m and m/q are integers, where c P means repeated addition c-times for point P. Now we can give a schematic pseudocode of ECPP using the notation of the previous theorem. ECPP(n) 1 E n random elliptic curve over over Z/nZ 2 m E n 3 if m = q s, where s is probably prime and the factorization of q is known 4 then goto 8 5 else goto if s n then goto 1 8 P random point from E n 9 if q P is not defined 10 then return COMPOSITE 11 if (m/q) P = O 12 then goto 8 13 if m P O 14 then return COMPOSITE 15 while we are not sure 16 do ECPP(s) During the application of the algorithm we get a strictly monotone decreasing series of numbers s, the first of which is n itself. Practically this algorithm terminates either in case of a composite n, or in a case when s is a small prime, which can be proved by easy effort. The effective proof for our 1002-digit candidate was carried out by the ECPP package version developed by F. Morain. This is a free program, which can be found on page [8]. The running time was sec on a processor AMD Athlon 64. References [1] Bondarenko, B. A., Pascal Triangles and Pyramids, Their Fractals, Graphs and Applications, The Fibonacci Association, Santa Clara, 1993, [2] Bressoud, D. M., Factorization and Primality Testing, Springer-Verlag, New York, 1989,

9 [3] Farkas, G., Kátai, I., Informatical Algorithms (Number Theory) (in Hungarian), ELTE Eötvös Kiadó, Budapest, 2005, [4] Freund, J. E., Restricted Occupancy Theory - A Generalization of Pascal's Triangle, Amer. Math. Monthly, 63 (1956), [5] Járai, A., Computatitonal Number Theory (in Hungarian), ELTE IK, Budapest, 1998, [6] Kallós, G., Generalizations of Pascal s Triangle (in Hungarian), MSc thesis, ELTE, Budapest, 1993, [7] Kallós, G., The Generalization of Pascal's Triangle from Algebraic Point of View, Acta Acad. Paed. Agriensis, XXIV, (1997), [8] Morain, F., The ECPP homepage, [9] Morton, R. L., Pascal's Triangle and Powers of 11, Math. Teacher, 57 (1964), [10] Sloane, N. J. A., On-line Encyclopedia of Integer Sequences, 117

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