From: however@alumni.caltech.edu (Everett W. Howe)
Subject: Re: consecutive power values of polynomials
Date: 18 Jan 01 19:05:39 GMT
Newsgroups: sci.math.numberthy
Summary: How many consecutive values of a polynomial can be square?

Hello all,

Gary Walsh asked:

> 1. Let p(x) be a nonsquare (monic) quadratic polynomial with integer
> coefficients. Is there an upper bound c for which f(0),f(1),...,f(c)
> can all be squares? I haven't yet found such a polynomial with
> f(0),f(1),f(2), and f(3) all being square.

I don't know about the upper bound c, but I can find an infinite
family of nonsquare f with f(i) a square for i = 0,1,2,3.
For example, x^2 + 492*x + 36 works.  Here's the construction:

Suppose a,b,c,d are integers.  It is easy to work out that the condition
that there be a polynomial f = x^2 + p*x + q in Z[x] such that
f(0) = a^2, f(1) = b^2, f(2) = c^2, and f(3) = d^2 is equivalent
to the conditions that
     (1)  a^2 + c^2 = 2*(b^2 + 1)
and  (2)  a^2 - 3*b^2 + 3*c^2 = d^2.

(The condition that f not be a square in Z[x] turns out to be
equivalent to b not being an element of {a+1,a-1,-a+1,-a-1}.)

There are many parametric solutions to (1).  One such solution is

    a = e - 1
    b = 5*e + 2
    c = 7*e + 3.

With these values for a,b,c, condition (2) becomes

   (2')  d^2 = 73*e^2 + 64*e + 16 = (8*e + 4)^2 + 9*e^2

There is an infinite family of solutions to (2').  For example,
suppose r and s are integers that satisfy

   (3)  (8*s + 3*r)^2 - 73*r^2 = 32

Then if we let e = r^2 - s^2 and d = 3*(r^2 + s^2) we get a
solution to (2').

Of course, there are infinitely many solution pairs to (3), and it's
easy to check that the polynomials f we get from them are almost
never squares.

The polynomial x^2 + 492*x + 36 corresponds to taking r = 2, s = -3.

 -- Everett

________________________________________________________________________
Everett Howe                          Center for Communications Research
however@alumni.caltech.edu                           4320 Westerra Court
http://alumni.caltech.edu/~however                  San Diego, CA  92121
==============================================================================

From: alf@ics.mq.edu.au (Alf van der Poorten)
Subject: Concerning:consecutive power values of polynomials
Date: 18 Jan 01 23:20:21 GMT
Newsgroups: sci.math.numberthy

Early references on some of the issues involved in Gary Walsh's question are

H.~Davenport, D.~J.~Lewis and
A. Schinzel, `Equations of the form $f(x)=g(y)$',{\it Quart. J.
Math.\/} {\bf12} (1961), 304--312.

H.~Davenport, D.~J.~Lewis and A.~Schinzel,
`Polynomials of certain special types', {\it Acta Arith.\/} {\bf9}
(1964), 107--116.

My quick (and no doubt careless) rereading suggests these provide a
guarantee of a bound $c(d,k)$, and rather more. In any case, for what
it's worth, I put up my hand with a confident answer of 'Yes!' to
Gary's questions. Interestingly, the case exemplified by Everett Howe
is, I think, one of the problem cases of Theorem 1 of the first paper.

One might want to look at quantitative versions of Hilbert's
Irreducibility Theorem. Apparently there are results of Dorge from
the 1920s (and I seem to recall somewhat more recent such results of
Mike Fried). Quite independently of Gary's question, the other day
Gerry Myerson alerted me to a paper of K R Matthews, MR:30 #1109, in
which it is shown that if a polynomial f is quite near to a $k$-th
power for almost all integer values $x$, $|f(x)-y^k|<\varepsilon x$,
then $f$ differs by a constant from a $k$-th power of a polynomial.

----------
At round about 23:46 -0500 17/01/2001, quelqu'un qui se fait passer
pour Gary Walsh  pretend m'avoir ecrit la chose suivante:
1. Let p(x) be a nonsquare (monic) quadratic polynomial with integer
coefficients. Is there an upper bound c for which f(0),f(1),...,f(c)
can all be squares? I haven't yet found such a polynomial with
f(0),f(1),f(2), and f(3) all being square.

2. More generally, if p(x) is an integer polynomial of degree d
and not a k-th power, is there an upper bound c in terms of d and k
for which f(0),f(1),...,f(c) can all be k-th powers?

--
------------------------
Alf vdP, alf@math.mq.edu.au
Department of Mathematics Macquarie University, Sydney 2109  Australia
phone: +61 2 9850 8947 fax: +61 2 9850 8114
home: +61 2 9416 6026   mobile: +61 4 1826 3129 (from MQ: #6335)