mersenneforum.org Sum of digits of x and powers of x
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 2022-11-07, 21:51 #12 mart_r     Dec 2008 you know...around... 15508 Posts Take it, it's free While this thread is still on the front page, let me drop this number here: Code: 977501592599998899778899999999987869899997877899798979989989676889999889999899775899989798999998789769998994897988599999999988989899879989699897999798898997599889876898998999994977897978999988999939998887 It's a 204-digit prime number whose digital sum (1651) is larger than the digital sum of its cube (1639). I found it less than a day after my last post in this thread.
 2023-02-27, 20:19 #13 mart_r     Dec 2008 you know...around... 23·109 Posts Fifth powers riddle This could make for a nice puzzle, or at least serve as inspiration for further endeavors: (assume all mentioned variables being positive integers) For any base b, are there only finitely many numbers x not divisible by b such that the sum of digits of x is larger than the sum of digits of $$x^5$$ (in base b)? Or is there a threshold $$b_0$$ above which all $$b > b_0$$ can have infinitely many (or at least one) such solutions? (Is $$b_0$$=283 for fifth powers?) Some solutions for $$b \leq 100$$ (searched up to $$x=10^8$$): Code:  b x 8 4* 27 9*, 23 32 2*, 4*, 8*, 16* 39 177716 40 20* 53 8210 54 18*, 36*, 138* 55 31 60 42 64 16*, 32*, 48*, 245408* 72 36* 77 822816 79 16255431 90 299047 92 52881676 96 24*, 48*, 6600*, 17256* 98 7140* * semi-trivial solutions, since b|x^5 For $$b > 100$$, more and more non-trivial solutions appear (the next one is b=102, x=4767). b=27 seems to yield the smallest non-trivial solution. Or might there be a smaller b for which such a solution can be found? What about higher prime powers p? It is trivial to show that there are arbitrarily large bases b for which $$x \geq 3$$ (mutually, all $$x^n$$ for $$1 \leq n \leq p-1$$) is a solution whenever $$b=x^p+2-x$$. Excluding all those trivial and semi-trivial solutions, for p=7 so far the smallest (in terms of b) non-trivial solution I found was b=492, x=121820. Is it possible to find a non-trivial solution for larger p? Have I overlooked a way to trivially construct solutions?

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