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 92 Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger 52 What is the role of mathematical intuition and common sense in questions of irrationality or transcendence of values of special functions? 25 Cool mathematics I can show to calculus students. 13 Conjectural closed-form representations of sums, products or integrals 7 Find what values of $n$ give $\varphi(n) = 10$

### Reputation (4,899)

 +20 Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger +10 Definition of a dominating function and the Dominated Convergence Theorem. +10 What is the role of mathematical intuition and common sense in questions of irrationality or transcendence of values of special functions? +10 Does there exist a bounded function of class $C^{\infty}$ such that all of its derivatives at $0$ coincide with the derivatives of $e^{-x} +x+1$?

### Questions (25)

 14 Necessary and sufficient conditions for a polynomial in $\mathbb{Z}[t]$ to have an $n$th root in $\mathbb{Z}[[t]]$ 10 Is there an intuitive proof of the identity $\sum_{L \subset S} \prod_{x \in L} (x-1) = \prod_{x \in S} x$ from general principles? 9 Prove that $\sum_{n=1}^\infty \mu(n) \log n/n =1$. 8 A field which is not algebraically closed but has no extensions of a fixed degree(s)? 8 Prove that a holonomic (p-recursive) difference equation returns only integral values

### Tags (97)

 134 number-theory × 24 52 special-functions 98 numerical-methods × 2 52 irrational-numbers 94 arithmetic × 2 52 philosophy 60 gamma-function × 4 27 sequences-and-series × 11 52 transcendental-numbers × 2 22 elementary-number-theory × 9

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 Mathematics 4,899 rep 1527