LIBRARY OF THE 
 
 UNIVERSITY OF ILLINOIS 
 
 AT URBANA-CHAMPAICN 
 
 510.8^ 
 ho. 6.I-&0 
 
 CO 
 
Digitized by the Internet Archive 
 
 in 2013 
 
 http://archive.org/details/generatingpseudo74farr 
 
UNIVERSITY OF ILLINOIS 
 GRADUATE COLLEGE 
 DIGITAL COMPUTER LABORATORY 
 
 REPORT NOo 74 
 GENERATING PSEUDO-RANDOM NUMBERS IN THE ILLIAC 
 
 By 
 
 Carl Co Farrington, Jr. 
 
 December 7 > 19 56 
 
 This work has been supported by 
 
 Contract N6ori-71 Task XXX 
 
 United States Navy 
 
 ONR NR Okh 001 
 
GENERATING PSEUDO-RANDOM NUMBERS IN THE ILLIAC 
 Carl C. Farrington, Jr. , University of Illinois 
 
 Introduction 
 
 This report contains results of research directed toward finding 
 a better method of generating pseudo-random numbers in the ILLIAC . When we 
 speak, loosely, of random numbers we mean numbers generated by a random 
 process. We shall call a process random if we cannot predict the next number 
 to be generated. 
 
 Since very long sequences of random numbers are required in Monte 
 Carlo computations, reading such numbers into the computer is not practical. 
 A possible alternative would be to generate inside the computer numbers by a 
 random physical process. This approach has the disadvantages that additional 
 equipment is required and that the computations are not reproducible . The 
 other alternative is to generate numbers by a determinate arithmetic process. 
 Of course, such numbers will not be "random." However, inasmuch as one is 
 only interested in obtaining trustworthy answers to problems, such "pseudo- 
 random" numbers may be satisfactory. Of course, whether the results of a 
 computation for which a particular sequence of pseudo-random numbers is used 
 should be given as much confidence as those for which a randomly selected 
 sequence of numbers is used, depends upon the particular problem. Accord- 
 ingly, the user must decide for himself how much confidence to place in the 
 numerical results he obtains. 
 
 These considerations led to placing primary emphasis on obtaining 
 a fast method of generating numbers, one taking only about 1/2 millisecond 
 per number, which generates a very long sequence. This is the sense in which 
 we speak of a "better" method. 
 
 For the most part this report contains the results of three tests on 
 two methods of generating pseudo-random numbers. These tests each determine 
 whether some particular property of the numbers is statistically predictable. 
 One of the methods tested is that used in the present library routine. The 
 other is a modification of a method proposed by Tocher . 
 
 Tdcher, K. D„, "The application of automatic computers to sampl- 
 ing experiments," J. Roy. Stat. Soc, ser. B, vol. 16, (1954) 39-61. 
 
I» Methods of generating pseudo- random numbers 
 
 The method used in the present library routine is the "middle-of- 
 
 (2) —19 —38 
 
 the-square" method^ . A certain 38 binary digit number (2 +3X2) 
 
 proposed by N. Metropolis is taken as the starting point. This number is 
 squared and the middle 38 digits of the product are used as the second mem- 
 ber of the sequence. This is squared to obtain the third member, and so on. 
 The first 1,000 numbers are usually discarded. The sequence is about 700,000 
 numbers long, after which it rapidly degenerates to a sequence of all zeros. 
 
 The first objective of this research was to find a faster method, 
 one taking on the order of one-half millisecond to generate one number. Ac- 
 cordingly, processes involving the use of multiplication or division orders 
 had to be ruled out. The method finally arrived at uses five starting numbers. 
 A linear combination of the last four is added to a number obtained by permut- 
 ing the digits of the first number, to obtain the 6th number of the sequence. 
 The same linear combination of the 3rd, 4th, 5th and 6th numbers is then added 
 to the permuted digits of the 2nd number to obtain the 7th number, and so on. 
 The linear combination used has only integral coefficients so the linear 
 combination can be formed by a series of additions, subtractions, and shifts. 
 The shifting operations are used simultaneously to permute the digits of the 
 other number. Secondly, the red tape is reduced by generating five numbers 
 instead of one each time the routine is entered. The overall result is that 
 the routine takes 3 milliseconds to generate five numbers, so that the num- 
 bers are generated at a rate of 0.6 millisecond per number. 
 
 The method proposed by Tocher was to form a linear combination of a 
 fixed number of the preceding numbers to obtain the next number in the se- 
 quence, without permuting the digits of any of them. He recommended that 
 the sequence of binary digits composing this sequence of numbers be paired 
 and that these pairs be replaced by single digits according to the following 
 code. 
 
 00 reject 
 
 01 
 
 10 1 
 
 11 reject 
 
 (2) 
 
 v 'Hammer, Preston C, "The Mid-Square Method of Generating Digits," 
 
 Proceedings of a Symposium July 29, 30 and July 1, 1949, National Bureau of 
 
 Standards Applied Mathematics Series 12. 
 
 -2- 
 
He demonstrated that this scheme would improve several types of non-random- 
 ness in the original sequence. He recommended that the resulting sequence 
 of digits be regrouped as variables and subjected to the pairing process again 
 to obtain the final sequence of random numbers. 
 
 Since the ILLIAC is a parallel machine, the time which would be con- 
 sumed in coding digits in the manner described would be absolutely prohibi- 
 tive for obtaining long sequences of numbers. 
 
 It was thought that a linear combination might be used directly as 
 a method of generating pseudo-random numbers without the use of the improve- 
 ment scheme suggested. One sequence, the Fibonacci numbers, which involves a 
 linear combination of only two of the preceding numbers is rather well known. 
 The Fibonacci numbers are obtained by making each number the sum of the two 
 preceding numbers, using 0, 1 as the starting numbers, so that one obtains 
 
 0, 1, 1, 2, 3, 5, ••■> The starting numbers are represented in the computer 
 
 -39 -39 
 
 as * 2 and 1 <* 2 . The sequence increases quite rapidly, and integers 
 
 larger than 2 are soon obtained. Overflow occurs and the sum must be re- 
 garded as a sum modulo 2 . The Fibonacci numbers and other sequences of 
 numbers obtained as linear coirbinations of preceding numbers have been found 
 to have a periodicity of their terminal digits which makes them unsuitable 
 as sources of random numbers. An analysis showing this result follows.* 
 We consider a sequence of integers modulo 2 given by 
 
 q 
 
 A A = <T a. A . , n > (l) 
 
 n+q , -^- , k n+q-k ' x 
 
 with starting numbers A„, A, , . . . , A _ . We assume that the last coeffic- 
 ient a is an odd integer. This is a sufficient condition for periodicity 
 
 of the terminal digits, for if a is odd it is relatively prime to any power 
 
 s 
 of 2. Hence for any s, a has an inverse modulo 2 , so that the equation 
 
 q - 1 
 k = 1 
 
 a A = A - > a, A 
 q n n+q *=3- 1 Tc n+q - k 
 
 5 
 
 has a unique solution modulo 2 for A in terms of its first q successors. 
 
 ^ n 
 
 It is clear that if the s terminal digits of q successive numbers in the 
 
 * The results and proof given here were suggested by those in an article by 
 Vern Hoggatt, "A type of periodicity for Fibonacci numbers," Mathematics 
 Magazine, vol. 28, no. 3, (1955), 139-142, analyzing the Fibonacci numbers 
 in their decimal representation. 
 
 -3- 
 
ii i 
 
 H i 
 
 
 : i 
 
 U ,il 
 
sequence are the same as the s terminal digits of q successive preceding num- 
 bers 
 
 T (A . ) = T (A . ) , i = 0, 1, ... , q - 1 (2) 
 
 s n + i + M' s n + i ' ' ' ' n v 
 
 where T (A) represents the s terminal binary digits of A, then the s terminal 
 s 
 
 digits will be periodic for i^ of period M. Since there are only a finite 
 number of possibilities for the s terminal digits of any number, the exis- 
 tence of such numbers n and M is assured. But if the coefficient a is odd, 
 then equation (2) will hold for all i^ -n due to the uniqueness of the pre- 
 decessor of any q numbers. Thus the periodicity of the terminal digits will 
 begin at the very beginning of the sequence. This result can be stated as 
 
 (R) Let A [be a sequence of integers modulo 2 given by (l) with a odd. 
 Then for any integer s $ 40 there exists an integer M such that 
 
 T (A, __ J = T (A. ) 
 s k + M s k 
 
 for all k. 
 
 If we denote the period of the s terminal binary digits by p we can 
 
 5 
 
 show now that 
 
 • «s - 1 
 
 P s ^ 2 Pl 
 
 To obtain this result we consider an auxiliary sequence 
 
 q 
 
 F = S" a, F , , n £ 0; 
 
 n + q . ~ , k n + q - k ' ' 
 
 1 q-2 ' q - 1 
 
 (The Fibonacci numbers, for which q = 2 and a, = a„ = 1 are a special case 
 of this) We shall show by induction that 
 
 i - 5" y a. , F _ .A , , n, p ^ 
 
 n+q+p k t- x p J+k p+q-l-j n+q-h » * 
 
 (3) 
 
 using the convention that a, = for k > q. For p = we have for the right 
 hand member 
 
 -4- 
 
ll I I 
 
 II " l" 
 I! I I" 
 
( X a, A . ) F . = A 
 v .-fi k n+q-k' q-1 n+q 
 
 by equation (l), since F , = 1, which holds for all n ^ 0. The assumption 
 that equation (3) holds for p = r and for all n f? yields 
 
 n+q+(r+l) (n+l) + q + r 
 
 q r 
 
 = *£ 2- a. , F , .A . . 
 
 k ti jT J k r+ ^- 1_ J n+1+ q~ k 
 
 q r 
 
 ' k?i w) a J +1+k "l r+q-(j+l) n+q-(k-l) 
 q-1 r+1 
 
 k tb jti a J +k ?r+q -J An+q " k 
 
 Using the convention that a, = for k > q we may write 
 
 q r+1 r+1 
 
 A / -^ = X ST a. . F .A , + zl a. F .A 
 
 n+q+(r+l) ^ jT J +k r+q-j n+q-k f^ j r+q-j n+q 
 
 q 
 
 ~\ A - F 
 
 f^ n+q-k r+q 
 
 If r +1 < q the second summation may be written 
 
 r+1 q q 
 
 <S a. F .A = S~a. F .A - ^ a. F .A 
 
 pj j r+q-j n+q -^ j r+q-j n+q _ ^ =r+2 j r+q-j n+q 
 
 q 
 
 = £"a. F . A 
 
 jr x J r+q-j n+q 
 
 for F = ... = F _ ? = 0. If r + 1 ^ q, since a. = for j > q, we again ob- 
 tain 
 
 r+1 
 
 < a.F .A =<a.F .A 
 
 >]_ J r+q-j n+q <^ j r+q-j n+q 
 
 -5- 
 
From the defining equation we have 
 
 r+1 q 
 
 y a. F .A = ( STa. F .) A = F A 
 
 TZ-y 3 r+q-j n+q f% j r+q-j' n+q r+q n+q 
 
 From equation (l) the third summation is 
 
 q 
 
 (T A A . ) F - A F 
 v /^, k n+q-k r+q n+q r+q 
 
 so we obtain 
 
 q r+1 
 
 A / n % = S" £ a. , F .A +F A -A F 
 
 n+q+(r+l) £. 7Z* J +k r+ 1~3 n+q-k r+q n+q n+q r+q 
 
 q r+1 
 j^i jTq J + ^ (r+l)+q-l-j n+q-k 
 
 which establishes the result for p = r+1. Hence it is true for all p, n 
 20. 
 
 If M is the period of the s terminal digits of the F' s, then there 
 are integers B. such that 
 
 F M+*i = V*' i = °' lj "•" > q " 2 
 
 F M . = B n 2 S + 1 
 M+q-1 q-1 
 
 Hence, since M must be greater than q-1, equation (3) implies 
 
 q q-1 
 
 F 
 2M+ 
 
 i ^ k jf a j + k F M+q-l-j F M+i-k 
 q-1 q 
 
 = j?o Si aj+k Fm+c i- 1 -J FM+i " k 
 
 q-1 q 
 = j| t= f +I a t F M+q-l-j F M+i+j-t 
 
 q-l-i q q-l-i j 
 
 K t^l^ FM+i+ 3-t F M+q-l-j " S ^ a t F M+i+j-t F M+q-l-j 
 
 q-1 q 
 
 a. F- . . . F„ 
 
 -q-l_j 
 
 5" < a. F„ . . F„ . 
 
 <=- . L 4- , t M+i+i-t M+q-1-; 
 
 j=q-i t=j+l 
 
 -6- 
 
II 1 1' 
 
 I 
 
q-l-i q-1 q 
 
 q-l-i j 
 
 = ^ F F +*> S~ a F F 
 
 .< n M+i+j M+q-l-j ■ • * TTi t M+i+j-t M+q-l-j 
 j=0 j i j J=q-i t=j+l J H J 
 
 j5_ t ? x ^ F M+i+j-t F M+q-l-j (4) 
 
 We note that for i < q - 1 the first summation is 
 
 q-l-i q-2 
 
 <"" F F = 2F F + ^1. F F 
 
 ^- Q M+i+j M+q-l-j " M+i M+q-1 21 r M+i+j M+q-l-j 
 
 g 
 
 Since each factor in the summation on the right is a multiple of 2 , the 
 
 2s > s+1 
 summati on is a multiple of 2 ^ 2 . The first term on the right is 
 
 2F M . F M , = 2B.2 S (B n 2 S +l) = B.2 S+1 + B.B -.2 2s+1 
 M+i M+q-1 l v q-1 l l q-1 
 
 Except for i = when the second summation in equation (4) is not present, 
 all factors in the second and third summations in equation (4) are multiples 
 of 2 so that these summations are multiples of 2 -5^2 . Hence we have 
 expressions of the form 
 
 s+1 
 F 2M+i = C i 2 for i = 0, 1, ... , q - 2 
 
 For i = q - 1 the third summation in equation (4) is not present, the second 
 
 s+1 
 is a multiple of 2 and the first is 
 
 F* . - B 2 ,2 Z3 .♦ B l2 S+1 ♦ 1 
 M+q-1 q-1 i q-1 
 
 which is of the form 
 
 s+1 
 F „ _ = C n 2 +1. 
 
 2M+q-l q-1 
 
 Hence for the auxiliary sequence 
 
 P s+1^ 2p s 
 and we notice that if m is a number such that 
 
 -7- 
 
F , - B -.2 m + 1, B - odd 
 
 P m + q-i q-i ' q-i 
 
 then 
 
 p =2 p , s ~ m 
 ^s m 
 
 Finally, we obtain our result by letting M be the period of the 
 s terminal digits of the F's. Then by equation (3) with n + q = i and 
 p = M 
 
 q q-i 
 
 hi+1 = /£-, A a j+k F M+q-l-j A i-k 
 k=l j=0 " n j 
 
 q-i q 
 = j?o t=f + i &t k± ^ '"^^ 
 
 ** q ^ ^a A F 
 
 = R, , A. + r C a, A. x . F M n « - 2 > t i+j-t M+q-l-i 
 
 M+q-1 1 /■= ,<4r t l+j-t M+q-l-j .*-. *-t ° id 
 
 n j=l t=l - j=l t=l 
 
 By our previous results this is of the form 
 
 A,, . = (B ,2 s + 1) A. + D2 S 
 Tt+i q-1 i 
 
 Hence 
 
 T s ( W " T s (A i ) ' 4 * ° 
 
 Thus the p for the A 1 s must be a divisor of the p for the F's for each s. 
 s ^s 
 
 The periods for the A' s need not be the same as for the F's. For example, 
 if the starting numbers for the A's are all even numbers then p, = 1. Con- 
 sequently our result should read 
 
 < s-l 
 
 P s (A) £ 2^ p x (F) 
 
 The periodicity just demonstrated of the terminal digits of the 
 numbers obtained by the linear combination method makes these digits too 
 predictable to be regarded as random. For this reason the method was modi- 
 fied to include permutation of the digits of one of the numbers of the 
 
combination. If (a Q , a., ... , a_g) denotes a forty binary digit number 
 the permutation operation 77" chosen is given by 
 
 IT C a Q* a -ij ••• y a 39' = ^ a 0' a L.* a 5 J *** 3 a 39* a l* a 2' a 3 
 
 With this modification of the linear combination method the condition for 
 periodicity of the s < 39 terminal digits is no longer met, for the s ter- 
 minal digits of a number no longer depend only on the s terminal digits of 
 the q preceding numbers. The combination investigated uses q = 5 an d is 
 given by 
 
 A =7A , + A -4A + 3A , + 7T A 
 n+5 n+4 n+3 n+2 "' n+1 n 
 
 This combination has the advantage that the equation 
 
 IT k = A r -?A ,-A _ + 4A ^ - 3A . 
 n n+5 n+4 n+3 n+2 •* n+1 
 
 has a unique solution for a member of the sequence in terms of its 5 immediate 
 
 successors, since the permutation operation If has a well-defined inverse. 
 
 The choice of the other coefficients and of q were rather arbitrary, except 
 
 that one case of a choice of q = 2 was found to be unsatisfactory. 
 
 In the machine the number A - is formed by first placing A in the 
 
 A register and A in the Q register. A left shift of one place is performed 
 
 to give 2 A . in the A register. To this is added - A _ and + A , giving 
 ° n+4 n+2 n+1 ° ° 
 
 2A ,-A _ + A ... Then a left shift of two places is performed, and to 
 n+4 n+2 n+1 r 
 
 the result is added -A ,+A „ - A n to give 4(2A ,-A _ + A , ) - A . 
 
 n+4 n+3 n+1 & v n+4 n+2 n+1 n+4 
 
 + A „ - A ,=7A ,+A _+3A ... In the meantime the digits q, , q^, 
 n+3 n+1 n+4 n+3 n+1 & ^1 J ^2 f 
 
 q„ of A have been added to the end of the number in the A register, and the 
 ^3 n 
 
 digits q. , q^t ... 9 q^g have been shifted left three places, the last three 
 places in the Q register being replaced by zeros. Adding the number in the 
 Q register to the number in the A register then completes the formation of 
 
 n+5 
 
 -9- 
 
II. Results of tests 
 
 We describe next the tests for randomness performed on the two meth- 
 ods of generating random numbers, the "iniddle-of-the-square" method and the 
 modified linear combination method, and the results of these tests. In the 
 main three tests were performed. The first was a chi-squared frequency test. 
 The binary digits of the numbers were grouped in fours , each group of four 
 being regarded as a sexadecimal digit. The test was a test of the frequencies 
 of the sexadecimal digits. For the middle-of-the-square method, since 38 
 digit numbers are produced, the first 36 binary digits of the first number 
 were used as the first 9 sexadecimal digit sj the last two binary digits of the 
 first number and the first two of the second number were used as the tenth 
 sexadecimal digit, and the remaining 36 binary digits of the second number were 
 used as the next nine sexadecimal digits. All tests of the middle-of-the- 
 square method began with the 1001st number of the sequence. Since the modi- 
 fied linear combination produces numbers 40 binary digits in length, 10 
 sexadecimal digits were obtained from each such number. 
 
 For the first 4,800,000 sexadecimal digits, representing 3A °f the 
 sequence, for the middle-of-the-square method the frequency test gave a value 
 of 16.6. The probability of getting a higher value of chi-squared is 35/&o 
 For the first 4,800,000 sexadecimal digits obtained by the modified linear 
 combination method a chi-squared value of 12.8 was obtained, which corresponds 
 to a probability of 62$. 
 
 The second test performed on the two sequences was a poker test. In this 
 test a "hand" of five sexadecimal digits is classified as a bust, one pair, 
 two pairs, three of a kind, full house, straight, four of a kind, or five of a 
 kind. Hands were tested in groups of 10,000. Testing every 5th group of 
 10,000 beginning with the first such group making a total of 280,000 hands a 
 chi-squared value of 13«81 was obtained for the middle-of-the-square method. 
 This value corresponds to a probability of 5°6$ of obtaining a higher value of 
 chi-squared. For 280,000 hands selected in the same manner a chi-squared value 
 of 7.46 was obtained for the sequence generated by the modified linear combi- 
 nation method. A chi-squared value of 7° 46 for seven degrees of freedom cor- 
 responds to a probability of 39/°. 
 
 -10- 
 
The third test applied is similar to a run or gap test for columns 
 of binary digits and made use of the logical product instruction of the computer, 
 The logical product of two numbers is formed and tested to determine whether the 
 resulting number contains all zeros. If the probability of any digit of a 40 
 digit number being zero is one-half, the probability of the logical product of 
 two numbers containing all zeros is (3/4) • If the logical product of the 
 first two numbers did not contain all zeros, the product of it with the third 
 number was formed. If this number was not all zeros its product with the fourth 
 number was formed, and so on. The numbers were divided into groups of 20 and a 
 determination was made of how many factors in the logical product of these num- 
 bers were required before a number containing all zeros was obtained. The 
 number of factors were classified as two or less, three, four, ... , twelve, 
 thirteen or more. The number of groups of 20 falling into each of these class- 
 ifications was compared to the theoretical distribution. The theoretical dis- 
 tribution was slightly different for the 38 digit numbers generated by the 
 middle-of-the-square method than for the 40 digit numbers generated by the modi- 
 fied linear combination method, of course. For the first 700,000 numbers of 
 the sequence obtained by the middle-of-the-square method a chi-squared value of 
 9.1 was obtained, which corresponds to a probability of 60$. For the first 
 1,000,000 numbers of the modified linear combination sequence chi-squared was 
 5.8, for which the corresponding probability is 89$« 
 
 This third test has the advantage that it is very fast, since opera- 
 tions are performed on whole numbers rather than on sexadecimal digits. For 
 example, testing the first 1,000,000 numbers of the sequence obtained by the 
 modified linear combination method required less than 30 minutes of computing 
 time. For this reason this test was the first applied to two other trial meth- 
 ods. One of these was a modified linear combination method which used only 
 two starting numbers. This was tried because it was felt that a routine which 
 generated only one number each time it was entered would be more convenient, 
 and this feature would not be too costly in time if only two numbers were used 
 in the combination. However, the chi-squared value obtained for the one se- 
 quence was 190.3, which is far beyond the acceptable range. 
 
 -11- 
 
The method of analysis applied to the linear combination method is 
 not applicable to the modified linear combination method, because the permu- 
 tation operation V does not have the necessary algebraic properties. Indeed, 
 there does not appear to be any promise that any method of analysis will yield 
 illuminating results. But perhaps this is not an altogether bad feature of a 
 method for generating random numbers. At any rate, in lieu of an analytic 
 determination of the length of the sequence the first ten million numbers of 
 the sequence were tested for repetition. It was found that the sequence does 
 not repeat within the first ten million numbers. As this test was being made 
 every 100,000th group of five numbers of the sequence was printed out, as an 
 aid to future users of the routine who might wish to check for possible mach- 
 ine error or wish to use different parts of the sequence. This list is given 
 in the appendix. 
 
 -12- 
 
APPENDIX 
 
 
 
 F085N08L+- 
 J25291+706 
 63F95019F7 
 1L6-479F+3 
 662487 -L5 6 
 
 500000 
 
 7350338N5+ 
 J108-4N+5- 
 L9NOL5456L 
 -L8N0J41N5 
 77L-77027J 
 
 3500000 
 
 9-2F917FN3 
 AN-JJ23L72 
 5L0F2J8JJ+ 
 -8N8FFF5N1 
 3J7068NL-9 
 
 4000000 
 
 F00-80L4+J 
 0LL58L6177 
 F5604564J7 
 43-2660+L3 
 
 90NL-0N49+ 
 
 7000000 
 
 9+F87-LL86 
 6-F-N15F9L 
 342+57+9NJ 
 2F3-F68N+5 
 417719F9N4 
 
 7500000 
 
 J928+NN+JF 
 6191261+J3 
 1J1N3412+L 
 068369FN-F 
 -32F9F203J 
 
 1000000 
 
 4500000 
 
 8000000 
 
 -2515+6-3F 
 674J-946-7 
 148+21JJ5+ 
 005-1LN5+8 
 4956J3918J 
 
 40JJ3J3+F- 
 +NF5F73779 
 F5F3888+87 
 64+7-921+1 
 N-JNF7613F 
 
 N57-L7-37N 
 35364J-N09 
 F7J129F6FL 
 F5+N3993J8 
 42704-+916 
 
 1500000 
 
 -J13L36L51 
 60360L269- 
 LJJ1082-F0 
 37473-U24 
 L99F6NL-79 
 
 5000000 
 
 NNLJJ263N1 
 8477-6J-5N 
 LNN7960L82 
 N05N19J272 
 0L13FN97F3 
 
 8500000 
 
 9N3LOFNON4 
 4+-5FL8950 
 21L+2L8LF4 
 1+0L14J277 
 F-N61715+1 
 
 2000000 
 
 45FNON57L9 
 6208J2N18- 
 N+N-2L2N77 
 L2NJ3+F0J2 
 4J-F7F8898 
 
 2500000 
 
 5JJ4+L9331 
 L3-4340+F+ 
 0-6498L3N3 
 +NLL34LLJN 
 +65024+N2L 
 
 3000000 
 
 5500000 
 
 F9F-913F+5 
 353N4^931N 
 08JL60J287 
 3519595J+7 
 -1471NF0J6 
 
 6000000 
 
 6L864FLO65 
 103N 688482 
 OU60-6L25 
 -608J1+12J 
 74+454469L 
 
 6500000 
 
 9000000 
 
 6411+668N8 
 F535273553 
 -495N5L987 
 519+N9+N70 
 09081N-9F5 
 
 9500000 
 
 +25F73+L+3 
 2J-1F4J3+9 
 9NF6719F45 
 J34106NN21 
 
 915777+34- 
 10000000 
 
 292+805182 
 
 87F39N8L31 
 8838F1F37+ 
 3181F7JJ44 
 NF698N+042 
 
 7982J-230- 
 4-3569F605 
 F3488-J030 
 -2219FF741 
 80LJ39-J57 
 
 L94-L3N79+ 
 -+59895367 
 9L2+6L+3L7 
 2385719LF4 
 52-7596J44