GETTING THE SUM OF TWO PERFECT SQUARES WITH AN

ALTERNATIVE METHOD IN

ADRIAN JAY C. GALES

AIJAN MATHEW C. BRAGAT

NADEERA N. KALIM

THIS MATHEMATICAL INVESTIGATION IS PRESENTED IN PARTIAL FULFILLMENT

DEPARTMENT OF EDUCATION

REGIONAL SCIENCE HIGH SCHOOL

MALASIGA, SAN ROQUE, ZAMBOANGA CITY

MARCH 2020
 ACKNOWLEDGMENT

The researchers would like to acknowledge and express their sincerest gratitude to the
following who in one way or the other had contributed to the success of this project, to:

 Their beloved parents, for extending their encouragement and moral support
 R.S.H.S principal, Mrs. Ma. Arlindela E. Ramirez and the faculty members for their
moral support and consideration to the Researcher’s hectic schedule.
 To Mrs. Mercelita Medallo and Mr. Eric Jude Yu-Can II, for their encouragement to the
researchers.
 To their beloved friends and the rest of 11-Pythagoras students for their moral support in
accomplishing this mathematical investigation.
 To all those who in one way or the other help pave the way for the success of this study.
 Above all, to the Almighty God who provided the researcher’s knowledge and
understanding, strength, guidance, protection, and inspiration as well as sending the Holy
Spirit, which granted the success of their research study.

i
 ABSTRACT

In number theory, the sum of two squares theorem says when an integer n > 1 can be
written as a sum of two squares, that is, when n = + . An integer greater than one can be written
as a sum of two squares if and only if its prime decomposition contains no prime congruent to 3
modulo 4 raised to an odd power. This study aims to find an alternative method in getting the
sum of two perfect squares. It is focused on using a derived formula or method to express the
number as the sum of two squares which was derived from the original and if it can be used by
other people on other mathematical concepts. The researchers used a derived formula which was
taken from the original formula in finding the sum of two squares. Using the new derived
formula, the researchers proved the new formula by using it on different sets of numbers. With
this, the derived formula, 2p= 2 + 2( , which was derived from the formula p = + by substituting
and with 2 + 2( . It was utilized by using sets of numbers and was proven to be applicable and be
used easily. The researchers also used mathematical induction for it to be further proven. By
doing two versions of Mathematical Induction, ( k + 1) and , which are the formulas you use for
Mathematical Induction, it has been proven by the researchers that the formulated formula
cannot be proven by Mathematical Induction The researchers concluded, utilizing a derived and
new version of the original formula in finding the sum of two perfect squares and have been
proven to get the same result that you will get from the original formula. The researchers have
shown that the derived formulated formula in finding the sum of two squares is usable and can
be used by students and/or in mathematical concepts since it has been shown to have accurate
results

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 TABLE OF CONTENT PAGE

Acknowledgment………………………………………………………………..i

Abstract ………………………………………………………………………… ii

Table of content …………………………………………………………………iii

Nomenclatures …………………………………………………………………..

I. Introduction…………………………………………………………. 1

II. Result and Discussions…………………………………………….... 3

III. Conclusion and recommendation…………………………………..5

References ………………………………………………………………………..6

iii
 INTRODUCTION

A. Background of the study

Number theory is a vast and fascinating field of mathematics, also called "higher
arithmetic," consisting of the study of the properties of whole numbers. Primes and prime
factorization are very important in number theory. Excellent introductions to number theory may
be found in Ore (1988) and Beiler (1966). The great difficulty in proving relatively simple results
in number theory prompted no less an authority than Gauss to remark that "it is just this which
gives the higher arithmetic that magical charm which has made it the favorite science of the
greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses
other parts of mathematics." Gauss, often known as the "prince of mathematics," called
mathematics the "queen of the sciences" and considered number theory the "queen of
mathematics" (Beiler 1966, Goldman 1997). In number theory, the sum of two squares theorem
says when an integer n > 1 can be written as a sum of two squares, that is, when n = + . An
integer greater than one can be written as a sum of two squares if and only if its prime
decomposition contains no prime congruent to 3 modulo 4 raised to an odd power. This theorem
supplements Fermat's theorem on sums of two squares which says when a prime number can be
written as a sum of two squares, in that it also covers the case for composite numbers. The
primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as
sums of two squares in the following ways: 5=, 13=, 17 =, 29=, 31=, 41=. On the other hand, the
primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be
expressed as the sum of two squares. This is the easier USING AN ALTERNATIVE METHOD
IN GETTING THE SUM OF TWO PERFECT SQUARES 5 part of the theorem, and follows
immediately from the observation that perfect all squares are congruent to 0 or 1 modulo 4. This
study aims to find other ways on finding the sum of two squares.

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B. Aims of the study

This study aims to find an alternative method in getting the number as the sum of two perfect
squares. Specifically, it aims to answer the following question:

a. What method or formula can be used in finding the sum of two perfect squares?
b. Can this new method be utilized in mathematical concepts? ( problems, discussions
solutions, etc.)

C. Significance of the study

The purpose of this study is to find an alternative method in getting the number as the sum
of two perfect squares to be taught to students who have interests in mathematics or for new
knowledge to be added in mathematics especially to those students whose forte is algebra.
The study focuses on finding other or new method in getting the number as the sum of two
perfect squares. And to find out if it is applicable to be used in some mathematical concepts as a
new technique.

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 RESULTS AND DISCUSSION

What method can be used in finding the sum of the two perfect squares?
A.Findings

Let (n + m) be any integer

Derived formula: 2p = (n + m) ^2 + 2(k + l) ^2

Derivation: p + a^2 + b^2

2(p) = 2(a^2 + b^2), :.Multiply both sides by 2 ;

2p = 2a^2 + b^2

2p = 2a^2 +2b^2, :. Assume a = (n + m) and b = (k + l);

= 2p =2 (n + m) ^2 + 2(k + l) ^2

B.Proving

An example can be shown by assuming value :. a = 4 and b = 2

Original formula :

p = a^2 + b^2

p = (4)^2 + (2)^2

p = 16 + 4 = 20

The researchers will now use the derived formula by having the same given: a = 4 and b
= 2 as : a = 4 = (2 + 2) and b = ( 1+ 1).

Derived formula : (n+ m) ^2 is the addends of

2p = 2(2 + 2) ^2 + 2(1 + 1) ^2 (k + l) ^2 is the addends of

2p = 32 + 8 ;

2p = 40 = p = 20
2 2
The researchers arrived to the same answer

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 The researchers need to know if this is true ;

Since p is given = 20, we need to find (n + m) which is a (n + m) = ?

(k + l) = 2 = (1 + 1)
The researchers need to arrive to (n + m) = 4 Using the derived formula
substitute p and (k + l)
Solving :
2p =2(n + m) ^2 + 2(n + l ) ^2 ; Multiply 2(20) and 2(2) ^2
2(20) = 2(n + m) ^2 + 2(1 + 1) ^2 Transpose the 8 to get the n + m
2(20) =2(n + m) ^2 + 2(2) ^2 subtracts 8 from 40 and divide by
2
2(20) = 2(n + m) ^2 + 8
40 - 8 = 2(n + m) ^2 Find the square root of both sides
to get your answer
32 = 2(n + m) ^2
2 2 ;

√ 16 = √ (n+ m)2 = (n + m) = 4
The researchers had proven that the result of the derived formula is true.

2. Can this new method be of use in mathematical concepts? (Problems, discussions, solutions,
etc.)

The new derived formula can be used in mathematical concept, since this formula has
been proven to be of use. It can now be introduced in mathematical discussions, problems,
solutions, etc.

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 CONCLUSION AND RECOMMENDATION

Conclusion

In the study of finding alternative method in getting the numbers as the sum of two
perfect squares, the researchers have discovered that not all prime number can be written as the
sum of two squares. The researchers have shown that the new formulated derived formula for
getting the sum of two perfect squares can only be used by students and in mathematical
concepts since it has been shown to have accurate, whether you use the original formula or the
new derived form.

Recommendation

1. The researchers recommend finding other investigatory methods to find an acceptable

2. The current study of the researchers may be used by other researchers who are seeking
to conduct the same study with the same ideas but with different aims and goals in the future.

3. Try different approach in getting the numbers as the sum of two perfect squares.

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 REFERENCES

Retrieved from http://mathworld.wolfram.com/NumberTheory.html David Christopher."A

Retrieved from https://doi.org/10.1016/j.disc.2015.12.002 Zagier, D. (1990), "A one-sentence

Retrieved from https://archive.org/details/ElementaryNumberTheory/page Websites: Weisstein,

Retrieved from https://doi.org/10.1080/00029890.1990.11995565

Online Book: Underwood Dudley (1978). Elementary Number Theory (2 ed.). W.H. Freeman
and Company.

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 ACKNOWLEDGMENT

The researchers would like to acknowledge and express their sincerest gratitude to the
following who in one way or the other had contributed to the success of this project, to:

 Their beloved parents, for extending their encouragement and moral support
 R.S.H.S principal, Mrs. Ma. Arlindela E. Ramirez and the faculty members for their
moral support and consideration to the Researcher’s hectic schedule.
 To Mrs. Mercelita Medallo and Mr. Eric Jude Yu-Can II, for their encouragement to the
researchers.
 To their beloved friends and the rest of 11-Pythagoras students for their moral support in
accomplishing this mathematical investigation.
 To all those who in one way or the other help pave the way for the success of this study.
 Above all, to the Almighty God who provided the researcher’s knowledge and
understanding, strength, guidance, protection, and inspiration as well as sending the Holy
Spirit, which granted the success of their research study.