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. Example: p=779131 q=913177 p*p=607045115161 q*q=833892233329 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 Kp/Kpp=8/3=2.666...  Q2. Find a prime p with larger Kp/Kpp than 2.666...

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

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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 ( http://www.worldofnumbers.com/index.html) 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.

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Hakan wrote:

Q1:
p=9744457
q=5747449
p*p=94954442224849
q*q=33033170007601

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

http://www.asahi-net.or.jp/~kc2h-msm/mathland/math02/math0210.htm

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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