In Chapters 2 and 3 we proved the irrationality of certain numbers *ξ*. The results of Chapters 4 and 5 can be considered as modest extensions. For example the finiteness of the number of (*x*, *y*) in Z^{2} with *x*(*x*^{2}−2*y*^{2}) = *y* is equivalent to the fact that for all large *η* in Z, the number *ξ* with *ξ*(*ξ*^{2} − 2*η*^{2}) = *η* is of degree 3 over Q; or the irreducibility of *X*^{4} − 2*η*^{3}*X* − *η* gives a number of degree 4 over Q.

In the present chapter we wish to go further and prove that certain *ξ* are transcendental over Q.

In Chapters 2 and 3 the irrationality of *ξ* was proved by first assuming *ξ* in Q, then by some method constructing a sequence of small rational numbers *ξ _{n}* ≠ 0 (

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