Palindromes

Palindromes for the HP-41

Overview

1°) Integer -> A Palindrome
2°) Integer = Sum of Palindromes

1°) Integer -> A Palindrome

-"PAL" takes a positive integer in X-register and returns a palindrome in X-register
-For example if X = 678 , "PAL" computes:

    678+876     = 1554
1554+4551   = 6105
6105+5016   = 11121
11121+12111 = 23232

Data Registers: R00 = temp
Flags: /
Subroutines: /

01 LBL "PAL"
02 LBL 00
03 STO 00
04 XEQ 10
05 RCL 00
06 X=Y?
07 GTO 02
08 +
09 VIEW X
10 GTO 00
11 LBL 10
12 0
13 X<>Y
14 LBL 01
15 RCL X
16 10
17 ST* T
18 ST/ Z
19 MOD
20 ST+ Z
21 RDN
22 INT
23 X#0?
24 GTO 01
25 X<>Y
26 RTN
27 LBL 02
28 CLD
29 END

( 48 bytes / SIZE 001 )

STACK	INPUT	OUTPUT
X	N > 0	Palindrome

Example: 678 XEQ "PAL" the HP41 displays the successive results ( 1554 6105 11121 23232 )

and finally returns X = 23232

Notes:

-The result is meaningless if the sum becomes larger than 9999999999
-So you could add something like E10 X<=Y? E^X X<>Y after line 08
-Thus, the HP41 will display OUT OF RANGE in this case

2°) Integer = Sum of Palindromes

-"SPAL" takes a positive integer in X-register and stores in R05-R06-..... palindromes p₁ , p₂ , .... such that N = p₁ + p₂ + ...

Data Registers: R00 to R04 = temp R05-R06-......-Ree = the palindromes whose sum = N
Flags: /
Subroutines: /

01 LBL "SPAL"
02 10
03 STO 04
04 4
05 STO 00
06 X<> Z
07 GTO 01
08 LBL 10
09 0
10 X<>Y
11 LBL 02
12 RCL X
13 RCL 04
14 ST* T
15 ST/ Z
16 MOD
17 ST+ Z

18 RDN
19 INT
20 X#0?
21 GTO 02
22 X<>Y
23 RTN
24 LBL 01
25 ISG 00
26 CLX
27 STO 01
28 XEQ 10
29 RCL 01
30 X=Y?
31 GTO 04
32 LOG
33 RCL 04
34 X<>Y

35 X=Y?
36 9.1
37 FRC
38 X=0?
39 GTO 03
40 X<> L
41 INT
42 RCL 01
43 RCL 01
44 1
45 R^
46 +
47 2
48 /
49 INT
50 10^X
51 STO 03

52 MOD
53 -
54 RCL 03
55 -
56 STO 02
57 LASTX
58 /
59 X=0?
60 9
61 XEQ 10
62 RCL 03
63 MOD
64 RCL 02
65 +
66 STO IND 00
67 RCL 01
68 X<>Y

69 -
70 GTO 01
71 LBL 03
72 RCL 01
73 1
74 -
75 STO IND 00
76 LASTX
77 ST+ 00
78 LBL 04
79 STO IND 00
80 RCL 00
81 .1
82 %
83 5
84 +
85 STO 00
86 END

( 119 bytes / SIZE var. )

STACK	INPUT	OUTPUT
X	N > 0	5.eee

Where 5.eee is the cotrol number of the registers containing the palindromes

Example: Express 9999999998 as a sum of palindromes

9999999998 XEQ "SPAL" >>>> 5.010 ---Execution time = 20s---

and we have in registers R05 to R10:

     R05 = 9999889999
     R06 =         108801
     R07 =             1001
     R08 =               181
     R09 =                   9
     R10 =                   7
---------------------------
   SUM = 9999999998

Notes:

-Lines 33 to 36 avoid an error when the number N is very large ( close to E10 )

-The method works well but it doesn't always give the best solution
-Cf reference [1] for better algorithms... which would require many bytes with an HP41 !

References:

[1] Javier Cilleruello , Florian Luca , Lewis Baxter - "Every Positive Integer is a Sum of Three Palindromes"
[2] William D. Banks - "Every Natural Number is the Sum of Forty-nine Palindromes"

hp41programs

Palindromes for the HP-41