I have found a nice application of the WolframAlpha website that might be used as a good promotion for making more visible that amazing answer engine developed by Wolfram Research.
We based our approach in the computation of the roots of two similar functions $H_{2}(x)$ and $H_{4}(x)$, but we decided to use the web site WolframAlpha for doing this mathematical computation and thus, these roots are easier to verify by everyone in the public or commercial versions.
Using this evidence, we showed that $H_{2}(x) < 0$ and $H_{4}(x) > 0$ for large enough real numbers $x$. We proved that
\[\sum_{p_{k} > p_{n}} H_{4}(p_{k}) \leq 0\]
under the assumption that the Twin Prime Conjecture is false for a large enough $n$th prime number $p_{n}$. However, this contradicts the results obtained by the web site WolframAlpha. Consequently, the Twin Prime Conjecture must be true.
Take into account that the following paper is accessible to a large audience of number theorists, including graduate students (check the references for WolframAlpha's results, please):
Article The Smallest Gap Between Primes
Could this computation be good enough for solving Twin Prime Conjecture?
Thanks in advance,
Frank
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