Let a be the unique real number that satisfies 1=a+a3.
Let S be any non-empty finite subset of the powers of a, i.e. S⊂{a1,a2,a3,…,}.
A ) Show that if the sum of elements of S is rational, then it is either 1 or 2.
B) Find two subsets which sum to 1 and 2 respectively.
C) Show that there are infinite such subsets.
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The polynomial P(x)=x3+x+1 is irreducible over integers by Crohn's criteria, since P(2)=11. This implies P(−x) is irreducible too. This basically means that a cannot be the root of either a linear or a quadratic equation with integer coefficients. This easily implies the same for rational coefficients.
We will state the above explicitly as Lemmas.
Lemma 1: a is irrational
Lemma 2: a cannot be a root of Ax2+Bx+C=0, where A,B,C are integers.
Note: we can prove the above lemmas in a more elementary fashion but will not do that here.
To prove A)
Notice that ∑∞n=1an=a1−a=aa3=1a2.
We can show that a>1√3 by using the fact that x3+x−1 is monotonic. Thus any finite sum of powers of a is <3.
Notice a2=1a−1. This allows us to make the claim that:
Proposition: For any integer n≥0, there exist integers An,Bn,Cn such that
an=An⋅a+Bna+Cn
Proof: Easily proven by induction, using a2=1a−1.
Thus the sum of any finite powers of a can be written as
A⋅a+Ba+C
for some integers A,B,C.
If that is rational, by Lemmas 1 and 2, we must have that A=B=0, and thus the sum must be C which is an integer.
Since 0<C<3, it must be either 1 or 2.
Part B)
1=a+a3
2=a+a2+a3+a4+a5+a6+a7
Part C)
If N=aj1+aj2+⋯+ajk
with jk being the largest, rewrite as
N=aj1+aj2+⋯+ajk(a+a3)=aj1+⋯+ajk−1+a1+jk+a3+jk
to get a representation with k+1 terms instead of k. Repeat to get infinite representations.