Problems & Puzzles: Puzzles

Puzzle 971. The cube of N such that...

Carlos Rivera asks this:

What is the smallest N value such that N^k has K times each digit of N?

Q1. For K=3. a) Find N if N is a natural integer. b) Find N if N is a prime number

Q2.  For K=4. a) Find N if N is a natural integer. b) Find N if N is a prime number

Please notice changes in the last 24 hours
 

Contributions came from Emmanuel Vantieghem, Simon Cavegn and Dmitry Kamenetsky

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Emmanuel wrote on Set 24, 2019

Q1. a
The smallest number is  87624375.
This solution can be found in the sequence  A114259  at the OEIS.

Q1. b
There cannot be a prime solution.  If  p  was such, then we would have : p^3 == 3 p (mod 9).
This would imply  p  to be divisible by 3.

Q2. a
The smallest solution is  5702631489  and can be found in  A114260.

Q2. b
There is no prime solution.  If  p  was such, then we would have : p^4 == 4 p (mod 9)
which implies  p^3 == 4 (mod 9), which is impossible (all cubes are 0, 1 or 8 mod 9) !  

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Simon wrote on Set 26, 2019

K=1 1^1=1
K=2 72576^2=5267275776 (Allowing duplicate digits: Four '7' in result)
K=2 406512^2=165252006144 (Without duplicate digits)
K=3 87624375^3=672782675854638427734375 (Allowing duplicate digits: Six '7' in result)
K=3 516473892^3=137766973511455269432948288 (Without duplicate digits)
K=4 5702631489^4=1057550783692741389295697108242363408641
K=5 961527834^5=821881685441327565743977956591832631269739424
K=6 7025869314^6=120281934463386157260042215510596389732740014997586987548736
 
K=1 Prime 2^1=2
K=2 Prime 49591523^2=2459319153459529 (Allowing duplicate digits)
K=2 Prime 71620589^2=5129508768706921 (Without duplicate digits)

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Dmitry wrote on Set 27, 2019

It seems that the answers are in these sequences:
 

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