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PEN Math problems from the collection "Problems in Elementary Number Theory"

Math problems from the collection "Problems in Elementary Number Theory"

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PEN Math problems from the collection "Problems in Elementary Number Theory"

Math problems from the collection "Problems in Elementary Number Theory"

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3615 posts

Peter

#1 h May 25, 2007, 12:25 AM • 1 Y

Y by Adventure10

One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.

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ZetaX

#2 h May 25, 2007, 12:25 AM • 2 Y

Y by Adventure10 and 1 other user

Let $a$ be the common difference and $an+b$ the sequence. If $ak+b=m^2$ for some $k,m$ , then $\infty$ many squares are given by $(na+m)^2=n^2a^2+2nam+m^2=(k+an^2+2mn)\cdot a + b$ .

This immeadiatly generalizes to any powers.

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TTsphn

#3 h Oct 28, 2007, 11:30 AM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Here is the general of this problem
Problem 1 If an arithmetic sequence contain infinite p th power of an integer if and only if contains infinite $p^{th}$ power of an integer .
proof (It is an fundamental of Polynomial congruent) because $x_n$ is an arithmetic sequence then
$x_{n + 1} = an + b$
Suppose exist an p th power of an integer in this sequence $a_{n_0 + 1} = k^p$
Imply that $k^p\equiv - b (\mod a)$
Consider the number $(k + ta)^p$ $t > 0$
Because $(k + ta )^p\equiv k^p\equiv - b (\mod a)$ so $(k + ta)^p$ is an term of the sequence .
So there are many p th power in this sequence.
Problem 2 If a arithmetic sequence contain an $m,n$ power the it contains mn power . Prove by congruence . where $\gcd(m,n)=1$

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