Problems & Puzzles:
Squares sharing no one single digit
idea suggested by
but at the same time subverting it, I have constructed the following
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
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.
example is this one:
p*p = 116696699999166169, Kpp=3
Find a prime p with larger Kp/Kpp than 2.666...
Contributions came from Emmanuel Vantieghem &
Hakan Summakoğlu, Claudio Meller & Carlos Rivera.
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.
& Carlos examined almost simultaneously the Hisanori's results of squares having only 3 digits,
Filtering these results for the primes, all the following ones have Kp=9 &
6325349321582201, 8124407769251, 8427200114569499,
So we could no
find any solution 10/3, that was our target.