Find-all-the-values-of-n-such-that-determinant-digit-sum-n-3-n-and-show-that-n-has-at-most-2-digits-digit-sum-abc-a-b-c-

Question Number 168885 by Rasheed.Sindhi last updated on 20/Apr/22

Find all the values of n such that :

\begin{matrix} digit - sum (n^{3}) = n \end{matrix}

and

show that n has at most 2 digits .

^{} digit - sum (\overset{―}{abc . .}) = a + b + c + \dots

Commented by botir last updated on 20/Apr/22

v e r y n i c e

Commented by Rasheed.Sindhi last updated on 20/Apr/22

T h a n X s i r!

Commented by MJS_new last updated on 20/Apr/22

n=0 and n=1 are solutions so we search for n∈N∧n>1

n = 0 and n = 1 are solutions

so we search for n \in N \land n > 1

Commented by MJS_new last updated on 20/Apr/22

0=0^3 1=1^3 512=8^3 4913=17^3 5832=18^3 17576=26^3 19683=27^3 no more solutions the sum of a 6−digit number is ≤54 ⇒ no possible solution >54

0 = 0^{3}

1 = 1^{3}

512 = 8^{3}

4913 = 17^{3}

5832 = 18^{3}

17576 = 26^{3}

19683 = 27^{3}

no more solutions

the sum of a 6 - digit number is ⩽ 54

\Rightarrow no possible solution > 54

Commented by MJS_new last updated on 21/Apr/22

for Σdigits(n^k )=n with k∈N^★ we have to try all n∈N^★ (0 and 1 are always solutions) with 1<n<z with z being the greatest zero of f(z)=9×(1+⌊klog_(10) n⌋)−n

for Σ digits (n^{k}) = n with k \in N^{★} we have to

try all n \in N^{★} (0 and 1 are always solutions)

with

1 < n < z

with z being the greatest zero of

f (z) = 9 \times (1 + ⌊ k \log_{10} n ⌋) - n

Commented by Rasheed.Sindhi last updated on 20/Apr/22

Great!

T han k s sir!

Commented by Rasheed.Sindhi last updated on 21/Apr/22

Σdigits(n^k )=(Σdigits(n))^k ? Sir my question was about Σdigits(n^k ).

Σ digits (n^{k}) = {(Σ digits (n))}^{k} ?

S i r m y q u e s t i o n w a s a b o u t

Σ digits (n^{k}) .

Commented by MJS_new last updated on 21/Apr/22

sorry {it}^{'} s a typo . I corrected it .

Commented by Rasheed.Sindhi last updated on 21/Apr/22

ThanX Sir! Are these formulas result of googling?

T h a n X S i r!

A r e t h e s e f o r m u l a s r e s u l t o f

g o o g l i n g ?

Commented by MJS_new last updated on 21/Apr/22

no. brainwork. I used a simple calculator and started with n=1, 2, 3, ... then I found that there must be a border because the max sum of k digits is 9k...

Commented by Rasheed.Sindhi last updated on 21/Apr/22

No doubt your calculator is simple but your brain is as complex as mathematics is. No doubt your formulae are creative work sir!

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