(41) Exponentiation & Residue Reduction

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Pages 341 & 344 from Number Theory in Science and Communication, Second Enlarged Edition, © Sprinler·Veriag Berlin Heidelberg 1984 and 1986, ISBN 978-3-662-22246-1 (eBoook)

"A. A Calculator Program for Exponentiation and Residue Reduction
We list here a program for calculating the least nonnegative remainder of aⁿ modulo m on a Hewlett-Packard 41C or 41CV pocket calculator. In other words, in our acute bracket notation, the calculator calculates
    
The 3 variables, the base a, the exponent n and the modulus m are entered into the calculator in that order.
    To call the program, which is labeled "AN", from storage, one presses
    GTO "AN" .
To calculate, say,
    {2³⁴⁰}₃₄₁ ,
one proceeds by pressing the following buttons:
    2
    ENTER
    340
    ENTER
    341
To start the program, one presses
    R/S
    After 2 seconds the HP 41 display shows in rapid succession
    8.
    6.
    4.
    2.
This is the binary decomposition of the exponent 340.

Check: 28 + 26 + 24 + 22 = 340. Check!

    Next, the computer will start the necessary repeated squarings and reductions modulo 341, which will take about 7 seconds. Then the display will show the end result, a 1 (without a decimal point!).
Thus,
    {2³⁴⁰)₃₄₁ = 1 .
In other words, 341 is a pseudoprime to the base 2.
    Similarly, after pressing 2, ENTER, 170, ENTER, 341, R/S, the display shows in succession:
    7.    5.    3.    1.        1
where that last display (the 1 without the decimal point) tells us that
    {2¹⁷⁰}₃₄₁ = 1 .
    Proceeding in the same manner, we find
    {2⁸⁵}₃₄₁ = 32 .
i.e., 341 is not a strong pseudoprime to the base 2. Also, by using 3 as a base we find
    {3³⁴⁰}₃₄₁ = 56 .
Thus, 341 is certainly not an absolute pseudoprime (Carmichael number).
    However, for the modulus 2821, we find
    {2²⁸²⁰}₂₈₂₁ = {3²⁸²⁰}₂₈₂₁ = 1 .
two of the many steps necessary to show that 2821 is an absolute pseudoprime or a Carmichael number
    Note that our little calculator with a limited accuracy and a 10-digit display, in calculating (3²⁸²⁰}₂₈₂₁, has coped with a number having 1346 decimal digits! This has been made possible by the frequent intermediate modulo reduction that the program employs.

                        Listing for "AN"                       
Comment                          Step    Code            
call program                       01       LBL "AN"
decimal point                     02       SF 29
store modulus                    03       STO 18
get exponent                     04       RDN
store exponent                  05       STO 17
get base                           06       RDN
store base                         07       STO 16
constant                           08       2
store 2                             09       STO 01
take logarithm                    10       LN
store log                           11       STO 15
display no fractions             12       FIX 0
constant                           13       1
store 1                             14       STO 14
constant                           15       0
store 0                             16       STO 02
subroutine for calculating     17       LBL 10
binary representation of n    18       RCL 17
                                       19       LN
                                       20       RCL 15
                                       21       /
add small constant to          22       0.000000001
avoid inaccurate rounding     23       +
                                       24       INT
                                       25       ENTER
                                       26       ST- IND 01
display the binary                27       PSE
exponents of n                   28       1
                                       29       ST+ 01
                                       30       RDN
                                       31       STO IND 01
                                       32       2
                                       33       x <> y
                                       34       y^x
                                       35       ST- 17
                                       36       RCL 17
binary representation           37       x = 0?
of n completed?                  38       GTO 11
                                       39       GTO 10
subroutine for executing       40       LBL 11
repeated squaring               41       RCL IND 01
                                       42       x = 0?
                                       43       GTO 12
                                       44       RCL16
                                       45       RCL 18
                                       46       2
                                       47       /
                                       48       x < > y
                                       49       x > y?
                                       50       XEQ 16
                                       51       x²
                                       52       RCL 18
take remainder                   53       MOD
modulo n                           54       STO 16
                                       55       1
                                       56       ST- IND 01
                                       57       GTO 11
subroutine for cal-              58       LBL 12
culating intermediate           59       RCL 16
squaring results                  60       ST* 14
                                       61       RCL 14
                                       62       RCL 18
                                       63       MOD
                                       64       STO 14
                                       65       1
                                       66       ST- 01
                                       67       RCL 01
                                       68       2
                                       69       x = y?
                                       70       GTO 13
                                       71       GTO 11
                                       72       LBL 16
                                       73       RCL 18
                                       74       -
                                       75       RTN
Subroutine for                    76       LBL 13
recalling and                      77       RCL 14
displaying residue                78       CF 29
display residue                    79       STOP
ready to start over              80       GTO "AN"
                                       81       RTN
                                       82       END"

BEST!
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