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for Programs the HP 28S These programs o er weak emulation of some of the most fundamental of the Turbo Pascal programs discussed in the preceding two sections. If the HP 28S is to be used extensively, then one would want to enlarge the collection to make it more comprehensive. It would also be desirable to have programs that employ binary integer variables (unsigned 64-bit integers). Their use would allow calculations up to 1018 . You may nd it convenient to place these programs in a user subdirectory called NOTHY, say. CRT Function Determines the intersection of two arithmetic progressions. That is, it expresses the simultaneous conditions x a1 (mod m1 ) , x a2 (mod m2 ) as a single congruence x a (mod m) , if the intersection is non-empty. If the intersection is empty then the message "No such number" is returned. . . . a1 m1 a2 m2 Restrictions Algorithm Listing Stack Action . . . a m |ai | < 1011 , 0 < mi < 1011 . If the intersection is non-empty but m > 1011 , then the message "Bad mod" is returned. Same as for the Turbo Pascal program of the same name. a1 m1 a2 m2 m2 M STO m1 a2 a1 m1 MOD DUP a1 STO - LNCN m1 * SETM m1 * a1 + M Calls Comments LNCN, MULT, SETM This program may be invoked k 1 times in order to nd the intersection of k given arithmetic progressions. 107 Programs for the HP 28S FAC Function Stack Action FACtors n by trial division. Thus n = p1 p2 . . . pr with p1 p2 . . . pr . . . . p1 p2 . . . pr . . . n Restrictions Algorithm Listing 0 < n < 1012 Trial division. After powers of 2 , 3 , and 5 have been removed, trial divisors are taken from the reduced residue classes modulo 30 . 2 TRIAL 1 + TRIAL 2 + WHILE DUP2 DUP * REPEAT TRIAL 4 + TRIAL 2 + TRIAL 4 + TRIAL 6 + END DROP DUP IF 1 == THEN 440 .25 BEEP TRIAL Rho If n > 1012 then the factorization given is erroneous. TRIAL 2 + TRIAL 4 + TRIAL 2 + TRIAL 2 + TRIAL 6 + DROP END Calls See Also Comments GCD Function Stack Action Calculates the Greatest Common Divisor g of two given integers b , c. If b = c = 0 then the error message "(0,0) undefined." is returned. . . . b c Restrictions 108 |b| < 1012 , |c| < 1012 Programs for the HP 28S . . . g Algorithm Listing Euclidean algorithm. bc b ABS b STO c ABS c STO IF b 0 == c 0 == AND THEN "(0,0) undefined." KILL END WHILE c 0 > REPEAT b c MOD c b STO c STO END b Comments If |b| > 1012 or |c| > 1012 then the value returned is likely to be incorrect. INVERSE Function Takes a from the stack and returns a such that 0 < a < m and aa 1 (mod m) , if such a number exists. Otherwise it returns the error message, "No such number" and kills the program. . . . a |a| < 1011 , 0 < m < 1011 Extended Euclidean Algorithm. a a 1 LNCN DROP LNCN, MULT The value of m is speci ed by using the program SETM. . . . a Stack Action Restrictions Algorithm Listing Calls Comments LNCN Function Given a LiNear CoNgruence ax b (mod m) , numbers x0 and m0 are found such that the solutions are precisely those x such that x x0 (mod m0 ) , if the congruence has a solution. Otherwise the error message "No such number" is returned, and the prgram is killed. 109 Programs for the HP 28S Stack Action . . . a b |a| < 1011 , |b| < 1011 , 0 < m < 1011 Extended Euclidean algorithm. . . . x0 m0 Restrictions Algorithm Listing ab b M MOD b STO M 1 0 0 c u0 u1 q WHILE c0> REPEAT a a c / FLOOR DUP q STO c * - c a STO c STO u0 u1 DUP u0 STO q * - u1 STO END b a / DUP DUP FLOOR IF > THEN DROP "No such number." KILL END M a / M m1 m2 m1 M STO IF u0 DUP 0 < THEN M + END MULT M m2 M STO Calls Comments MULT The value of m is speci ed by using the program SETM. 110 Programs for the HP 28S M Function Restrictions Comments A stored variable, used as the modulus in congruence arithmetic. 0 < M < 1011 The value of M is speci ed by the program SETM. MULT Function Stack Action Given a and b , returns the unique number c such that c ab (mod m) and 0 c < m. . . . a b Restrictions Algorithm Listing 0 a m , 0 b m , 0 < m < 1011 Russian multiplication base g = [5 1011 /m] . See Problem 21 on p. 83. of the text. xy 0 5E11 M / FLOOR s g WHILE y 0 REPEAT yyg/ g*-x s STO END s . . . c > FLOOR DUP y STO * s + M MOD x g * M MOD x STO P? Function Stack Action . . . m Programs for the HP 28S Determines whether a number m is prime. . . . m is prime. is composite. is too large. 111 Restrictions Algorithm 1 m 25 109 Strong pseudoprime tests are performed until m is proved to be prime, or until m is found to be composite. Use is made of the known fact that if 1 m 25 109 and if m is a strong probable prime base 2, 3, 5, 7 then m is prime unless m = 3,215,031,751 . m IF m 25000000000 > THEN m "is too large." ELSE m SETM 2 SPSP IF 0 == THEN m "is composite." ELSE IF m 2047 < THEN m "is prime." ELSE 3 SPSP IF 0 == THEN M "is composite." ELSE IF m 1373653 < THEN m "is prime." ELSE 5 SPSP IF 0 == THEN m "is composite." ELSE IF m 25326001 < THEN m "is prime." ELSE Programs for the HP 28S Listing 112 7 SPSP IF 0 == m 3215031751 == OR THEN m "is composite." ELSE m "is prime." END END END END END END END END 526 .25 BEEP Calls MULT, POWER, SETM, SPSP POWER Function Stack Action Finds the unique number a such that 0 a < m and a xk . . . x k Restrictions Algorithm Listing |x| < 1012 , 0 k < 1012 , 0 < m < 1011 The exponent k is expanded in binary, and the base x is repeatedly squared. xy 1 p x M MOD x STO WHILE y 0 > REPEAT y y 2 / FLOOR DUP y STO 2 * IF 1 == THEN p x MULT p STO END x x MULT x STO END p 113 (mod m) . . . . a Programs for the HP 28S Calls Comments MULT The value of m is speci ed by the program SETM. QUIT Function Listing Returns the machine to the root directory. HOME RHO Function The Pollard RHO method is used to search for factors of m , say m = ab. The numbers a, b found are not necessarily prime. This method should only be applied to numbers m that are already known to be composite. . . . m a b Stack Action . . . m Restrictions Algorithm 0 < m < 1011 The sequence un (mod m) is generated by the recurrence un+1 u2 + n s (mod m) , with the initial condition u0 = r . The program takes r = s = 1. The quantity (u2n un , m) is calculated for successive values of n , until it takes a value strictly between 1 and m. If it takes the value m then the program is terminated with the message "Test inconclusive." In such a case one should edit the program to use a di erent value of s. However, one should avoid s = 0, s = 2 . m 11 rs m DUP SETM r DUP 1 WHILE DUP 1 == REPEAT DROP DUP MULT s + SWAP DUP MULT s + DUP MULT s + SWAP DUP2 - m GCD END 3 ROLLD DROP2 DUP IF Programs for the HP 28S Listing 114 m == THEN "Test inconclusive." ELSE DUP2 / END 880 .25 BEEP Calls MULT, SETM SETM Function Stack Action Assigns a value to the variable M , used as the modulus in congruence arithmetic. . . . M Restrictions Listing 0 < M < 1011 DUP DUP DUP IF 0 > SWAP 1E11 < AND SWAP DUP FLOOR == AND THEN M STO 0 FIX ELSE "BAD MOD" KILL END . . . SPSP Function The Strong PSeudoPrime test base a is applied to the number m . If it is found that m is a strong probable prime base a then the number 1 is returned. Otherwise m is proved to be composite, and the number 0 is returned. . . . a 0 . . . or 1 115 Stack Action Programs for the HP 28S Restrictions Algorithm Listing 1 < m < 1011 The strong pseudoprime test, as explained on p. 78 of the text. a IF M DUP DUP 2 / FLOOR 2 * == SWAP 2 > AND THEN 0 ELSE 00 jk M1WHILE DUP DUP 2 / FLOOR DUP 3 ROLLD 2 * == REPEAT SWAP DROP 1 k + k STO END DROP a SWAP POWER DUP DUP IF 1 == SWAP M 1 - == OR THEN DROP 1 ELSE 1 CF WHILE 1 FC? j k 1 - < AND REPEAT DUP MULT 1 j + j STO IF DUP 1 == THEN 1 SF DROP 0 END IF DUP M 1 - == THEN 1 SF DROP 1 END END IF 1 FC?C THEN DROP 0 END END END Calls Comments MULT, POWER The value of m is speci ed by using the program SETM. 116 Programs for the HP 28S TRIAL Function Stack Action This is a subroutine used in the program FAC. It executes TRIAL divisions, so that the integer k is found such that dk n . . . . d . . . d n/dk d . . . n d Restrictions Listing 0 < d < 1012 , 0 < n < 1012 WHILE DUP2 MOD 0 == REPEAT DUP 3 ROLLD / OVER END k Programs for the HP 28S 117 118 Programs for the HP 28S Further Resources The sources listed below have not been fully tested. They may contain bugs, and should be used at your own risk. All the software listed below can be obtained by using the le transfer program ftp. If you have the software ftp and a link to the internet, then you may connect to a remote host and retrieve software. To do this, type ftp hostname and then press the Return (or Enter) key. Here hostname is the electronic address of the remote host. Alternatively, you may type ftp which will give you the prompt ftp>, and then type open hostname If you are prompted to provide a login name, then type anonymous Otherwise, type login anonymous Then you will be prompted for a password. Type your own electronic address. In most cases, the software you want is in a subdirectory. To change to the subdirectory pub , for example, type cd pub or else cd /pub You may have to change directories yet again, say to the subdirectory msdos . You can accomplish both changes in one step by typing cd /pub/msdos If, in exploring subdirectories, you nd that you have wandered too far down the directory tree, and want to return to the parent of the current directory, then type cd .. Further Resources 119 To ascertain the name of the current working directory on the remote machine, type pwd In any particular directory, you can obtain a list of the les and subdirectories by typing ls or else dir These commands are handy if you are unsure of the exact name of the le you wish to retrieve. Once you are in the appropriate subdirectory, and have determined the name of the le you wish to transfer, say clint1.tex , type get clint1.tex If you wish to get all the les in the current directory, you can ask for a multiple get by typing mget * Text les can be transferred in the default ASCII mode, but most software must be transferred in binary. To prepare for binary transfers, issue the command type binary When you wish to terminate your ftp session, type quit Since the availability of software may change with time, the information provided below will gradually become inaccurate. If your system has the ARCHIE program, you may use it to locate sources of software. Try typing archie pari CLINT This lab manual, and the Turbo Pascal programs that are meant to be used with it, can be obtained by anonymous ftp from ftp.math.lsa.umich.edu in the subdirectory /pub/clint. The following les are available: FILENAME readme clint0.tex 120 CONTENTS general instructions title page of lab manual Further Resources clint1.tex clint2.tex clint3.tex clint4.tex error5-1.tex tpprgms2.exe forematter (pages ii x) of lab manual the labs (pages 1 68) program documentation (pages 69 106) programs for HP 28S, Further Resources Known errors in Niven-Zuckerman-Montgomery (Fifth Edition, First Printing) Turbo Pascal programs (archived) The .tex les will run on any system equipped with the TEX software, but the Turbo Pascal programs run only under DOS on IBM-compatible PCs. Both source code and compiled code are provided. If you plan to alter the source code or write your own programs you will need the Borland Turbo Pascal compiler, or else some other Pascal compiler. The le tpprgms2.exe is a self-extracting archive le that compresses into 1.3M what would otherwise be 145 Turbo Pascal les occupying 2.37M. The command type binary must be given before this le is transferred. The other les should be transferred as ascii les (issue the command type ascii). Instructions for decompressing tpprgms2.exe are found in the readme le. Users are encouraged to edit the laboratories, and to customize the manual for their own use. If you are unable to obtain this material by ftp, write to Mathematics Editor John Wiley & Sons 605 Third Avenue New York, NY 10158-0012 Upon receiving your request, you will be sent a printed copy of this lab manual and a 3.5 inch disk containing the tpprgms2.exe and readme les. For instructors only, the publisher will also provide a printed copy of the Solutions Manual for the Niven-ZuckermanMontgomery text. GP GP is an interactive calculator based on the PARI library of routines. See PARI for more details. Malm Donald E. G. Malm has written a large number of instructive programs, some in UBASIC, others in Turbo Pascal. These les can be obtained by anonymous ftp. Connect to the address vela.acs.oakland.edu, and then change to the subdirectory /pub/numbdroid. These are text les, so transfers should be made with the type ascii. If you have di culty obtaining these les, send an email message to malm@argo.acs.oakland.edu, or write to Professor Donald E. G. Malm Department of Mathematical Sciences Oakland University Rochester, MI 48309-4401 Further Resources 121 Numbers Numbers (Version 2.01) is a program written in Turbo Pascal 5.0 for IBM-compatible PCs by Ivo D ntsch of the Mathematics Department, Universit t Osnabruck, Moorlandstr. u a 59, 4500 Osnabruck, Germany. It o ers a wide variety of routines, accessed by a menu hierarchy. The program is aimed at students rather than researchers, and is available free of charge for educational purposes by anonymous ftp from clione.t2.uni-osnabrueck.de in the directory publ/msdos/math. For further information see the review in the AMS Notices 39 (1992), 840 841, by Louis D. Grey. PARI PARI takes its name from Pascal arithmetic, although eventually the code was written in C and assembler. It does not o er symbolic manipulation to the extent of Maple, for example, but it provides powerful routines for use in number-theoretic investigations. It was designed by C. Batut, D. Bernardi, H. Cohen and M. Olivier, with researchers in mind. It runs on the SUN3, SPARC, Macintosh II, NeXT, Amiga, on 32-bit PCs running MSDOS or OS2. It is available by anonymous ftp from megrez.ceremab.u-bordeaux.fr and also from math.ucla.edu, in the subdirectory /pub/pari. The transfer must be made in binary. Version 1.38, released in September, 1993, di ers from the former Version 1.37 in several important ways, most notably in the introduction of a package for computations in algebraic number elds. The package includes TEX les that produce a 130 page manual and a 30 tutorial. For more information, see the review of Q. Gouv a, AMS Notices 38 e (1991), 903 904. SEQUENCES N. J. A. Sloane s book A Handbook of Integer Sequences, Academic Press, New York, 1973 can now be queried by email. To submit sequences for examination, send an email message to sequences@research.att.com. Each line of your message should be of the following form: lookup 4 9 16 25 36 lookup 3 5 8 13 21 If your message is not of this form, then in reply you will recevie an explanation of the correct notation. A second program is also available, for the purpose of trying to identify a given sequence. To use it, send an email message to superseeker@research.att.com. Only one request may be made per message; your message should consist of a single line of the form lookup 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 122 Further Resources Start at the beginning, and give lots of terms. Negative entries are acceptable. The program will try very hard to nd an explanation. The program invoked is complicated, and still under development. Because some serious computing is executed, users are limited to one query per hour. If you encounter a bug with the program, please report it to Neil Sloane at njas@research.att.com. SIMATH SIMATH is written in C, and o ers many C-functions over algebraic structures such as arbitrarily long integers, rational numbers, oating point numbers, polynomials, Galois elds, matrices, elliptic curves, algebraic number elds, congruence arithmetic, etc. The package includes an interactive calculator, SIMCALC, that can access many of these routines, including arithmetic of nite elds and elliptic curves. In Version 3.8.1, released in March of 1994, includes an algorithm for calculating Groebner bases. The package runs successfully on a variety of machines, such as Sun 3/140 under SunOS 3.4, and Sun SPARCstation under SunOS 4.1.1. The package is available by anonymous ftp from ftp.math.uni-sb.de and from ftp.math.orst.edu UBASIC UBASIC is a high-precision version of BASIC for IBM-compatible PCs, written by Yuji Kida. It provides fast arithmetic for integers of up to 2600 digits, and is aimed at students of number theory. It comes with on-line help and sample programs. It is available free of charge by anonymous ftp from math.mps.ohio-state.edu in the subdirectory /pub/msdos/ubasic. The transfer must be made in binary. The main les are archived as .zip les. A suitable unzip program is provided in the same directory. The transfer must be made in binary. For further information see the reviews of Walter D. Neumann in the AMS Notices 36 (1989), 557 559; 38 (1991), 196 197. Version 8.74 was released in May, 1994. If you are using Version 8.30 or earlier and DOS5 or later then you should upgrade to avoid a memory-handling problem. Further Resources 123

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