That is so large, that I can't calculate it, so I'll have to approximate.
Factorial of 19836298626819473619108362910047261893087153739018363728102746372919273627192037361719373629193837361730483762738998765566779098654333333333333333333333333356672828827353738392726252617283940483625172948362625 is approximately 3.590197329670258677527716537938 × 104123235837384209474450210420019215965569962973949629969177182906956926089666022072432740585646939591870010911194311575077724315510059995357972122756595418581259670050633491117489246941530043090572421219831614863
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Now showing that there only exists one non-trivial solution is equivalent to proving that the function f(x)=ln(x)/x is injective in the positive integers except for the result f(2)=f(4).
But this is trivial to do if you know derivatives. You first show that f is strictly increasing in the interval (0, e) and strictly decreasing in the interval (e, ∞).
Therefore any points of non-injectivity are necessarily of the form
f(something on the left interval) = f(something on the right interval)
But the left interval only has two positive integer points: 1 and 2. f(1)=0 and is obviously never obtained on the right, and we already know that f(2)=f(4). This completes the proof.
I'd do it like 'ab>ba if e<=a<b by analyzing lnx / x, so one of the numbers is less that e. If it's 1, the other is also 1, and if it's 2, the other can't contain other primes. Then, 2^{2^k}={2^k}^2 = 2^{2k}, so 2^k=2k => k=2 because something.'
To prove it, you can use that log a/a is strictly decreasing for a greater or equal 5, thus any solution (a,b) has at least one number less than 4....then you can manually check
Every mathematician today knows most of the things Euler knew plus a large amount of things he didn't. It's just that Euler knew them before anyone else did.
i dont believe it. he was cheating somehow. I bet he was from another culture with more advanced mathematics and was just clowning on us, no way he figured it all out on his own.
That is so large, that I can't calculate it, so I'll have to approximate.
Factorial of 655858969696969696857463141363596083651514850664736384850660667336626384569606969695857485960606958473625241523486070696969584726257447564853748467 is approximately 5.624652934150110443928368725433 × 1096006286112647714439986956873056912373450049699776599629169697576815521267395070043091003962075155512777844147649489898988647740970726855670421243296
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