Monday, 14 February 2022

Squares that Parrot their Roots

I never noticed this pattern before today, not sure how it never came up. 

Look at the number 36.  You already know by memory it is the square of its last digit.  

How about 7,588,043,387,109,376, do you recognize the square roots of this 16 digit number.  Wouldn't it be cool if you could read it off the last half of the digits of the square, 87,109,376.  Check!

There is a direct route to get from 6 to this number, and more.  

The pattern seems to work like this:  6^2 =36.   Since six is a one digit number, we look at the first two digits of its square, but we change the leading digit, 3,  to its ten's compliment, 7.  Now if we square 76 we get, 5776...with the 76 right there as the caboose.

Ok so the 76 is preceded by a 7.  replace that with its ten's compliment, 3.  376 is the next number in our sequence, and 376^2 = 141,376. And if I understand correctly there is no number between 76 and 376 that produces this complete repetition.  

Can you find the next?  Take 1376 and make it 9376, and the square is 87,909,376.   

Now the next one has a clinker. If we took 09,376 we really don't have an additional digit, so you have to rake one more, 909,376.  You know what to do, the ten's compliment of the 9 is 1 and 109376^2 = 11,963,109,376.  

There is one more before you get to the 16 digit number I began with.  Now can you find the one after that?  You already know five of the six digits from the last one.

And Why?  All I know is this pattern seems to work up to some audaciously big numbers,,,(Thank you, Wolfram Alpha)

787109376^2=6.19541169787109376 × 10^17


81787109376^2 =6689131260081787109376

Uh Oh!!! double zero???? 


Not a proof of anything, but..... 

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