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It is known that there are no four square numbers in arithmetic progression. The same goes for cube numbers, powers of four, etc. Does it follow then that there is no quadratic polynomial with integer coefficients that contain an arithmetic progression of four (or more)? Same question for cubic polynomials etc...

Adam Rubinson
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    @Macavity Can a parabola intersect a line even three times? Yet there are three squares of integers in arithmetic progression. – David K Nov 11 '24 at 17:08
  • Could you clarify what you mean by a polynomial "containing" an arithmetic progression? Do you mean (say) that the numbers $f(0)$, $f(1)$, $f(2)$ form an AP in that order? Or perhaps that the there exist integer arguments $x_1$, $x_2$ etc so that the numbers $f\left(x_i\right)$ form an AP? It looks like other comments/answers interpret this in different ways – Chris Lewis Nov 11 '24 at 19:11

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For quadratic polynomials

$\{P(x)=ax^2+bx+c:a,b,c,x\in\mathbb{Z}\}$

there are none with four values in arithmetic progression. For if there were, we would also have

$4aP+(b^2-4ac)=4a^2x^2+4abx+b^2=(2ax+b)^2$

with four values in arithmetic progression, and four integer squares in arithmetic progression do not exist.

With cubic polynomials a set of four values in arithmetic progression is trivial because we can specify four function values independently, as for instance:

$P(0)=0,P(1)=9,P(2)=6,P(3)=3\implies P(x)=2x^3-12x^2+19x$

in which $P(4)=12$ and thus this cubic polynomial gives five values in arithmetic progression!

Reinhard Meier
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Oscar Lanzi
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For fourth power polynomials this is quite possible: in fact for any four integers in arithmetic progression there exists a fourth power polynomial $f(x)$ such that, when $x$ takes the value of any of those integers, $f(x)=x$.

If $p,q,r,s$ are integers in arithmetic progression, we simply put:

$$f(x)=(x-p)(x-q)(x-r)(x-s)+x$$

Similarly for $n$th power polynomials and $n$ integers in arithmetic progression.

Adam Bailey
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  • We are better than this. There are cubic polynomials over the integers with five values in arithmetic progression. – Oscar Lanzi Nov 15 '24 at 20:05