## Sunday, October 10, 2021

### n = a^m + b

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)

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

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