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Auxiliary Polynomials in Number Theory by David Masser

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1

Prologue

In this chapter we give a couple of examples where the method of auxiliary polynomials is used for problems that have no diophantine character. Thus we are not following Sam Goldwyn’s advice to start with an earthquake and work up to a climax.

Here is maybe one of the simplest examples.

There is an old chestnut which often turns up in problem-solving sessions: given a polynomial F in a variable X, can one always multiply it by a non-zero polynomial to get a product involving only powers Xp for p prime?

For example with F = X100 + 1 we have X3F = X103 + X3. But what about F = X100 + X3? Here multiplying by some P = aXd will not do. However

(X111X14)F = X11(X100X3)F = X11(X200X6) = X211X17.

At first sight it appears to be ...

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