Yes, in fact you cracked it. What I thought was notable about this solution was: - It arises naturally from a common length conversion. - We know that 0.3937 = 0.3937 1/2.54 = 0.3937007874... Given that the numbers themselves are so close, it would stand to reason that their geometric mean would be near their arithmetic mean, so that sqrt(0.3937 * (1/2.54)) .= (0.3937+(1/2.54))/2 = 0.3937003937... Both accurate to the decimals shown. This gives us a square root with a pretty repeated start block, which you observed and reproduced in another example. - But then there is the nice common factor between the block values 3937 = 31*127 254 = 2*127 This arises from the unusual square factor in the prime factorization: 999998 = 2*31*127^2 This has the peculiar side benefit that the quotient of the repeating blocks is a terminating decimal: 3937/254 = 15.5 Going back to our geometric mean identity, we get sqrt(0.3937 * (1/2.54)) = sqrt(0.155) = 0.3937003937... Which gives us a pretty square root of a terminating decimal. Multiplying by 20 on both sides then gives sqrt(62) = 7.874 007874 ... I will guess that small integer square roots with pretty blocks like this are rare in any base. - And an aside about the Mersenne primes You can't miss that 31 and 127 are Mersenne primes, which at first might seem incidental. However, consider that 2^20 = 1048576 .= 10^6 is close to a power of 10. Obviously this is well known, since we call 2^10 bytes a kilobyte, 2^20 bytes a megabyte, etc. We can break this up to 2 * 2^5 * (2^7)^2 = 1048576 .= 10^6 And if we tweak a couple of these factors slightly, we shouldn't affect the product too much, so lets tweak 2^5 and 2^7 down a notch: 2 * (2^5-1) * (2^7-1)^2 .= 999998 .= 10^6 So perhaps it is now less surprising that Mersenne primes might multiply to a value near 10^6, since Mersenne primes are slightly smaller than powers of 2 and 2^20 is slightly larger than 10^6. But of course, this is a hindsight analysis, since I knew the factorization to begin with. On 2/5/2012 3:09 AM, Robert Munafo wrote:
I don't see how the Mersenne factors matter.
The first two of David's properties, 1/2.54 = 00.3937 007874... and 1/39.37=00.0254 000508... are explained by their product being 99.9998 = (10^N-2) / 10^K, as I suggested before.
The other of David's properties, regarding the geometric mean, also comes from (10^N-2) / 10^K, as you can see here:
1/sqrt(99.9998) = .100000100000150000250000437500787501443752681255027353246111 1/sqrt(99.998) = .100001000015000250004375078751443776813002743871274179225808 1/sqrt(99.98) = .100010001500250043757876444018175282935398352327750882087668 1/sqrt(99.8) = .100100150250438288946436286857920839193514369563735147583946
These all have the same pattern, related to the series expansion of sqrt(1/(1-2x)), which you can get by "Series[sqrt(1/(1-2*x)),{x,0,8}]" in Mathematica or Wolfram Alpha. The coefficients are: 1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, ... The numerators are https://oeis.org/A001790, which are also the numerators of the sqrt(1/(1-x)) series expansion because only the denominators change.