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Lets use a Super-Calculator ! \(e^{\pi\sqrt{163}}=\) Integer ?

To see if you qualify :- \[(\text{Irrational})^{\text{Irrational}}=\text{Irrational} \\ \text{A. True} \quad \text{B. False} \quad \text{C. Unpredictable !}\]

Ok. Fine ! Now what will be your reaction if you come out to know that \(e^{\pi\sqrt{163}}\) is an integer. (waiting for your GIFs about your reaction). Don't you feel it strange that three irrational numbers \(e\) , \(\pi\) and \(\sqrt{163}\) can be combined to form an integer ?


In fact, the Indian Mathematician Srinivasa Ramanujan (1888-1920) first conjectured that \(e^{\pi\sqrt{163}}\) was an integer. He felt he found its value to be : \(262537412640768743.999.......\)


In 1972, computers had carried it out to 2 million places of 9's, but to be an integer one must know it repeats forever.

So, if 9 repeats forever, then \(e^{\pi\sqrt{163}}\) will be equal to \(262537412640768744\)

Now let me know if you're enough convinced to try to figure out its exact value !!!


Leave your amazing comments & ideas here. and Keep Exploring the Universe of Mathematics.

Note by Sandeep Bhardwaj
1 year, 7 months ago

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I am astonishment! I NEVER knew that \(\displaystyle { e }^{ \pi \sqrt { 163 } }\) could be an integer. I have so much to learn about mathematics. This is a perfect example of how simple intuition leads you astray. Can't trust your intuition, ya gotta do the math and work out all those decimals after the \(...8743.999...\)

Meanwhile, no, it's not true that \({ \left( Irrational \right) }^{ Irrational }=Irrational\) is always true. To see why, let \(x\) be an unknown. Then we solve for \(x\) the equation \({ x }^{ x }=2\quad \) or any other such rational number besides \(2\). Then in nearly all cases, \(x\) is not rational.

For a fascinating insight into this factoid, look up "Heegner Numbers" and check out Ramanujan's contribution to it. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin

I am astonishment! I NEVER knew that eπ163√ could be an integer.

Is this sarcasm? Because it isn't an integer and Poe's law. Siddhartha Srivastava · 1 year, 7 months ago

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@Siddhartha Srivastava Siddhartha, that was pretty cool, this is the first time I've heard of Poe's Law. It's right on target! Let me offer a corollary to Poe's Law, which is that no matter how outrageous or silly a politician's or pundit's statements may be, there will always be lots of people that believe them wholeheartedly.

Still, Heegner's numbers, to me, is a pretty fascinating subject. Heegner Number

Also here too Heegner Number

I really like the lineup of this very finite sequence of nine Heegner numbers

\(1, 2, 3, 7, 11, 19, 43, 67, 163\) Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin What's the proof that there are only 9 of them? Kartik Sharma · 1 year, 7 months ago

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@Kartik Sharma Look up Stark-Heegner Theorem What's interesting is that Carl Gauss as far back in the 19th century had conjectured that there would only be a finite number of them. But it took about a century to prove it, and another half century for other mathematicians to finally accept Heegner's 1952 proof of it, after it was upgraded by Stark in 1969. This has had a long history with a number of luminaries having worked on it, almost like Fermat's Last Theorem. Michael Mendrin · 1 year, 7 months ago

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@Michael Mendrin That's interesting! Kartik Sharma · 1 year, 7 months ago

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Wow!! Akshat Sharda · 1 year, 5 months ago

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Very interesting Arunesh Kumar · 1 year, 7 months ago

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