Problems & Puzzles: Puzzles

Puzzle 588. Squares sharing no one single digit

Following an idea suggested by Claudio Meller but at the same time subverting it, I have constructed the following puzzle:

Find pairs of primes p & q such that each one is an anagram of the digits of the other, and at the same time p^2 & q^2 share no one single digit.



Q1. Send your largest example

I also played a bit this week with a similar puzzle:

Find primes p composed by Kp distinct digits, its square p*p composed by Kpp distinct digits, such that Kp/Kpp is maximal.

My largest example is this one:

p=341608987, Kp=8
p*p = 116696699999166169, Kpp=3

 Q2. Find a prime p with larger Kp/Kpp than 2.666...


Contributions came from Emmanuel Vantieghem & Hakan Summakoğlu, Claudio Meller & Carlos Rivera.


Emmanuel wrote:

Concerning Q1, I found many solutions.  This is the last one :

 (p,q) =  (389291299,991928923), squares : (151547715477107401,983922988283939929).
If in addition we would ask for pandigital squares, I found :
 (p,q) = (991326827,187923269), squares : (982728877929887929,35315155031646361).

Concerning Q2.  This was much harder.  I could not find a better result of my own, so I visited Patrick De Geest's wonderfull World! Of Numbers ( where I found among interresting squares with only three different digits that the prime p = 8427200114569499  has square  71017701771000177071770101111001.  So, here we have : Kp = 9  and  Kpp= 3.


Hakan wrote:



Claudio & Carlos examined almost simultaneously the Hisanori's results of squares having only 3 digits, here:

Filtering these results for the primes, all the following ones have Kp=9 & Kpp=3

6325349321582201, 8124407769251, 8427200114569499, 999997323321167445187.

So we could no find any solution 10/3, that was our target.


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