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# transcend - Pade Approximations and the Transcendence of...

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Pade Approximations and the Transcendence of π Ernie Croot March 9, 2007 1 Introduction Lindemann proved the following theorem, which implies that π is transcen- dental: Theorem 1 Suppose that α 1 , ..., α k are non-zero algebraic numbers, and that β 1 , ..., β k are distinct algebraic numbers. Then, α 1 e β 1 + · · · + α k e β k 6 = 0 . The reason that this implies that π is transcendental is that if π were algebraic, then so is , which would mean 0 = e + 1 6 = 0 . We will not prove this general result (of Lindemann), but will instead show only that e α can never equal - 1 for any algebraic number α , which proves π is transcendental because e πi = - 1. The proof for the general case uses similar ideas to this special case. 2 The Proof 2.1 The Idea: Pade Approximations We begin by recalling the standard proof that e is irrational: Suppose e is rational. Then, n ! e must be an integer for all n sufficiently large; however, n ! e = I n + 1 n + 1 + 1 ( n + 1)( n + 2) + 1 ( n + 1)( n + 2)( n + 3) + · · · , 1

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where I n is the integer 2 n ! + n ! 2! + n ! 3! + · · · + 1 . It is easy to see that for n sufficiently large, n ! e = I + δ , where δ (0 , 1), which contradicts the fact that n ! e is an integer; so, e must have been irrational. Actually, what this proof gives us is the even stronger fact that there exists an infinite sequences of integers f n and g n tending to infinity, such that f n e - g n 0 . Indeed, just take f n = n ! and g n = I n . This brings us to the following basic fact: Fact. If α is some (possibly complex) number for which there exist sequences of integers f n , g n → ∞ such that f n α - g n 0 , and f n α - g n 6 = 0 , then α is irrational. 1 To show that e α , α is a non-zero rational, is irrational we will find such f n and g n . First, we begin with the case where α is an integer. If we can show this, then it follows that for any rational a/b we have e a/b is irrational (on taking b th powers). Our sequence of f n ’s and g n ’s comes from what are called Pade approxi- mations to e x . Basically, a Pade approximation is a pair of polynomials f ( x ) and g ( x ) such that e x f ( x ) g ( x ) for x near 0. There are methods for finding such good pairs f and g , and the simplest is to just use linear algebra. Basically, we try to find f ( x ) and g ( x ) of degree n so that the Taylor expansion of g ( x ) e x - f ( x ) 1 The proof is obvious, since if α = a/b were rational, then | a/b - g n /f n | = 0 or is at least 1 /bf n . Multiplying through by f n gives the result. 2
about x = 0 begins c 2 n +1 x 2 n +1 + c 2 n +2 x 2 n +2 + · · · This uniquely determines f and g up to scalar multiples. Such g and f can be found using the “pade” command in Maple; for example, Maple gives that in the case where g and f have degree 4, e x 1 + x 2 + 3 x 2 28 + x 3 84 + x 4 1680 1 - x 2 + 3 x 2 28 - x 3 84 + x 4 1680 . Notice here that g ( x ) = f ( - x ). This follows since if g ( x ) e x - f ( x ) has order of vanishing 2 n + 1 at x = 0, then so does e - x ( g ( x ) e x - f ( x )) = g ( x ) - e - x f ( x ) .

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transcend - Pade Approximations and the Transcendence of...

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