Mersenne primes, apart from 2^2 - 1 , have the form 2 a^2 - 1 , (as do all "Mersenne numbers", M_n = 2^n - 1 , when n is odd). The product of two numbers of the form 2 a^2 - 1 does not necessarily have the same form, although it will be of the form 2 x^2 - y^2 . Consider the Diophantine equation (2 a^2 - 1)(2 b^2 - 1)^2 = (2 c^2 - 1) . This has the "obvious" solutions where b = 0, 1, -1 and a = +/-c . It also has the not-so-obvious solutions where b = 2a , and c = a (8 a^2 - 3) . Take a = 4 in the not-so-obvious family, to get b = 8 and c = 500 , which gives David's factorization.

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.