n = a^m + b
Calculators: Swiss Micros DM41X, HP-32S/32SII, Casio fx-9750 GIII
n = a^2 + b
The following program breaks down a positive integer n into n = a^2 + b, where a is √n rounded to the nearest integer in attempt to minimize abs(b).
Examples:
800 = 28^2 + 16
662 = 26^2 - 14
525 = 23^2 - 4
This technique is useful, especially with the HP 32S and HP 32SII. Normally, real numbers take up 9.5 bytes of memory on the HP 32S series. Integers 0 - 100 only consume 1.5 bytes of memory for the original HP 32S and the range is extended to 0 - 254 for the HP 32SII.
When the program terminates, b is on the X stack while a is on the Y stack.
HP 41C/Swiss Micros DM41X Program: SQPLUS
01 LBL^T SQPLUS
02 FIX 0
03 ENTER
04 SQRT
05 RND
06 X^2
07 -
08 LASTX
09 SQRT
10 X<>Y
11 FIX 4
12 RTN
HP 32S and HP 32SII Program: SQPLUS
S01 LBL S
S02 FIX 0
S03 ENTER
S04 SQRT
S05 RND
S06 x^2
S07 -
S08 LASTx
S09 SQRT
S10 x<>y
S11 ALL
S12 RTN
Example: 684
Result:
Y: 26
X: 8
(26^2 + 8 = 684)
Additional Powers
Now let's extend the problem to include higher powers.
n = a^m + b
We can estimate a by:
n ≈ a^m
ln n ≈ ln(a^m)
ln n ≈ m * ln a
ln n / m ≈ ln a
a ≈ e^(ln n / m)
Round a to the nearest integer. Use this a to find b.
n = a^m + b
b = n - a^m
The script brkpower.py (225 bytes) calculates a and b for powers m = 2 to 6. This script was created on the Casio fx-9750GIII.
Casio fx-9750GIII Micro python Script: brkpower.py
# 2021-07-17 ews
from math import *
print("n = a**m + b")
n=float(input("n? "))
m=2.0
while m<7:
a=int(exp(log(n)/m)+.5)
b=n-a**m
s="{:.0f} = {:.0f}**{:.0f} + {:.of}".format(n,a,m,b)
print(s)
m+=1
Example: n = 860
Results:
860 = 29**2 + 19 (860 = 29^2 + 19)
860 = 10**3 + -140 (860 = 10^3 - 140, and so on)
860 = 5**4 + 235
860 = 4**5 + -164
860 = 3**6 + 131
Eddie
All original content copyright, © 2011-2021. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
