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# HP Prime and HP 41C/DM 41L: Sum of Two Squares

HP Prime and HP 41C/DM 41L:  Sum of Two Squares

Introduction

Given a positive integer n, can we find two non-negative integers x and y such that:

n = x^2 + y^2

(x and y can be 0, n is assumed to be greater than 0)

There are several theorems and lemmas that are connected to this famous problem.  As a point of interest, I will briefly describe them here.

1.  n does not have a representation (n can’t be written as x^2 + y^2) if any of n’s prime factors is congruent to 3 mod 4 and is raised to an odd power.

2.  If n has a representation, then for an integer k, k^2*n also has a representation.

3.  If n is prime and congruent to 1 mod 4, then n has a representation.  (n has the form of n = 4w + 1 for some non-negative integer w).

The program presented here is the use of iterations to find all possible pairs which fit n = x^2 + y^2.   Some integers do not have representations, others have more than one.  The program will show all possible combinations.

HP Prime Program SUM2SQ

EXPORT SUM2SQ(n)
BEGIN
// EWS 2019-07-21
// breaking n into a sum of 2 squares
LOCAL r,j,k,l;
// we can more than 1 representation
r:=IP((n/2)^0.5);
l:={};
FOR j FROM 0 TO r DO
k:=(n-j^2)^0.5;
IF FP(k)==0 THEN
l:=CONCAT(l,
{STRING(j)+”^2 + “+
STRING(k)+”^2 = “+
STRING(n)});
END;
END;

RETURN l;
END;

HP 41C/DM 41L Program SUMSQRS

Registers  used:
R00 = n
R01 = counter
R02 = temporary

01 LBL T^SUMSQRS
02 FIX 0
03 STO 00
04  2
05  /
06  SQRT
07  INT
08  1000
09  /
10  STO 01
11  LBL 00
12  RCL 00
13  RCL 01
14  INT
15  X↑2
16  –
17  SQRT
18  STO 02
19  FRC
20  X=0?
21  GTO 01
22 GTO 02
23 LBL 01
24 RCL 01
25 INT
26 T^X =
27 ARCL X
28  AVIEW
29  STOP
30  RCL 02
31  T^Y =
32 ARCL X
33 AVIEW
34 STOP
35 LBL 02
36  ISG 01
37  GTO 00
38  T^END
39  VIEW
40  FIX 4
41  RTN

Examples

Example 1:  n = 325
325 = 1^2 + 18^2
325 = 6^2 + 17^2
325 = 10^2 + 15^2

Example 2:  n = 530
530 = 1^2 + 23^2
530 = 13^2 + 19^2

Source:

Dudley, Underwood.  “Elementary Number Theory”  2nd Ed. Dover Publications: New York.  1978. ISBN 978-0-486-46931-7

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