Page 20 - Abstract book - TJSSF-2020

Thailand – Japan Student Science Fair 2020 (TJ-SSF 2020)
“Seeding Innovations through Fostering Thailand – Japan Youth Friendship”

Proving the product of four consecutive natural numbers is not a perfect

square by greatest common divisor and modulo
Witchaya Wittayasetthakun , Naphat Promprasert
1
1
Advisors: Supamit Wiriyakulopast , Nopporn Thamrongrat
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1

Princess Chulabhorn Science High School Nakhon Si Thammarat
1
Walailak Univerisity
2

Abstract

In this mathematical project we study the product of four consecutive positive integer numbers is not a
perfect square . We separate proof in two cases are (n - 1 , n + 2) = 1 and (n - 1 , n + 2n) = 3 where n
2
2
2
2
∈ Z for n ≥ 2. In case (n - 1 , n + 2) = 1 we use n ≡ 0,1 (mod 4) to solve. In case (n -1 , n + 2) = 3 is
+
2
2
2
2
2
divided into two cases, for gcd (p, 9p + 18p + 11p +2) = 1 we use n ≡0,1,4,9(mod16) to solve and case
3
2
2
gcd (p, 9p + 18p + 11p + 2) = 2 is proved by comparative inequality. So, we can conclude that the
3
2
product of four consecutive positive integer numbers is not a perfect square.

Keywords: Non-perfect square

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