• Title/Summary/Keyword: greatest common divisors

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  • Shin, Yong-Su
    • Journal of applied mathematics & informatics
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    • v.26 no.5_6
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    • pp.1215-1219
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    • 2008
  • We find a necessary and sufficient condition that the prime factors of $A^m+1$ and $A^n+1$ coincide for odd positive integers $n>m{\geq}1$. Moreover, we also find a necessary and sufficient condition that the set of all prime factors of $A^m+1$ is a subset of those of $A^n+1$ for $n>m{\geq}1$.

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Hong Jung Ha's Number Theory (홍정하(洪正夏)의 수론(數論))

  • Hong, Sung-Sa;Hong, Young-Hee;Kim, Chang-Il
    • Journal for History of Mathematics
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    • v.24 no.4
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    • pp.1-6
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    • 2011
  • We investigate a method to find the least common multiples of numbers in the mathematics book GuIlJib(구일집(九一集), 1724) written by the greatest mathematician Hong Jung Ha(홍정하(洪正夏), 1684~?) in Chosun dynasty and then show his achievement on Number Theory. He first noticed that for the greatest common divisor d and the least common multiple l of two natural numbers a, b, l = $a\frac{b}{d}$ = $b\frac{a}{d}$ and $\frac{a}{d}$, $\frac{b}{d}$ are relatively prime and then obtained that for natural numbers $a_1,\;a_2,{\ldots},a_n$, their greatest common divisor D and least common multiple L, $\frac{ai}{D}$($1{\leq}i{\leq}n$) are relatively prime and there are relatively prime numbers $c_i(1{\leq}i{\leq}n)$ with L = $a_ic_i(1{\leq}i{\leq}n)$. The result is one of the most prominent mathematical results Number Theory in Chosun dynasty. The purpose of this paper is to show a process for Hong Jung Ha to capture and reveal a mathematical structure in the theory.