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Saturday, 6 July 2024

Hidden Pandigitals

When we think of pandigital numbers, it is a number like 1263480759 that comes to mind. Each of the digits from 0 to 9 occur exactly once. There are 3,265,920 such numbers (leading zeroes not being considered) out of the total of 3,486,784,401 possible ten digit numbers. That's a representation of about 0.094%.

There are however, other ways in which pandigital numbers can arise and one way is in the decimal approximations of the square roots of whole numbers when approximated to ten digits NOT ten decimal places. Some of the digits will occur in the whole number part of the square root and the rest will occur in the decimal part. An example is the square root of 1362 where we have:1362=36.90528417

Note that the final digit arises from truncation of the infinite decimal and not from rounding. Essentially, the decimal point is ignored and so associated with certain whole numbers are their pandigital square roots expressed as whole numbers themselves with the decimal points ignored. The complete list of such numbers, up to 40000, is as follows:136236905284171843429301758625405039841267428065421708935507742091638568968304215796146011208345976151431230568974175471324650897183931356207948203371426078539227101506983742235601534926708258871608943752274871657920384307281752940386322861796830542326151805962347331441820549367344991857390642371941928574603
Of course it's easy enough to see where the decimal point should be. These numbers form OEIS A113507 (permalink). We can extend this idea to cube roots and in so doing the first number to make an appearance is 2017 because:(2017)1/3=12.63480759
The numbers up to 40000 are (permalink):201712634807593053145069328799502150837964151392473806519155332495083671183572637954108242142893047156244242901386574314573156742089326543196284750396053408657291
These numbers form OEIS
 A119517. This approach can be extended to fourth roots and beyond. We can also consider whole numbers raised to let's say 1/π. For example:36381/π=13.59746028
jUpt to 40000, the following whole numbers lead to pandigitals when raised to the power 1/π:3638135974602871091682940375102711892064735115721965273840138182079431586144352108547963205392359047168209812375089614262202549731608271582578419036273132583094176350222795816403353302803619574359012817964035370232845703691
Not surprisingly this sequence of numbers is not found in the OEIS.

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