Let's Solve A Cool Factorial Equation

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FACTORION

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[robot]

I like how you everytime don't fail to surprise us with new cool idea,
Big fan of you keep it up 😊

[rickostidich]

It's 145 in the usual (boring) base 10, but we also have 144 in base 5, 661 and 662 in base 15, 725 in base 27, and 166 in base 35, for example. 🤠

[mikecaetano]

Off the top it appears that 8! would play a similar role in the five digit case as the role played by 5! in the three digit case.

[rickostidich]

Regarding the subject of this video, that is a 3-digit "factorion" in whatever base, I already wrote a comment with the list of canonical numbers satisfying this equation for all the bases. In addition to those: if we allow digits greater than or equal to the number base (why not?! - I use them very often for several useful reasons) we also have 133₂, 224₃, 304₃, 441₃, 442₃, 540₅ (=5∙5²+4∙5¹+0∙5⁰ =5!+4!+0!); if we allow a leading 0 (only 1 is needed here) we also have 011₂, 010₃, 013₅, 043₇, 034₉, 014₂₂, 045₃₅ (=4∙35+5 =0!+4!+5!), and many more in greater bases; if we allow both previous cases concurrently we also have 012₂, 020₂, 023₃.

[migtrewornan]

Mathematical notation 'abc' usually means the product of 'a', 'b' and 'c' and there's nothing in the question which says it means concatenation 'a*b*c is a three digit integer' does not imply concatenation.

[PeterKasting]

0! = 1, not 0. You noted this correctly several times in the second half of the video, but in the first half treated it as 0 a couple of times.

[jayktomaszewski]

he made some mistakes with his arithmetic. 5! + 5! + 0! = 241 & 5! + 0! + 0! = 122