Wikipedia:Reference desk/Archives/Mathematics/2014 August 3

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August 3

Perimeter of the USA

Has the perimeter of the 48 continental states of the USA been calculated? Is there an "official" value? Thanks. Joseph A. Spadaro (talk) 06:56, 3 August 2014 (UTC)

It can't be done. See Coastline paradox. HiLo48 (talk) 07:26, 3 August 2014 (UTC)

And yet we have List of countries by length of coastline and List of countries and territories by land borders! These won't get you all the way to the answer, because you'll have to disentangle the lower 48 from the total, but the question can be answered in practical terms. RomanSpa (talk) 12:12, 3 August 2014 (UTC)

"In practical terms" usually means choosing a limit on how small a kink must be to avoid being ignored. Unfortunately, this resolution is rarely mentioned, even though it is assumed in and significantly affects the measurement. I suppose cartographers have some default that they work to. —Quondum 16:25, 3 August 2014 (UTC)

For seacoasts I imagine using the shortest path lying between average low– and high-tide contours. The land boundaries of the US are defined as straight lines between markers. —Tamfang (talk) 22:56, 4 August 2014 (UTC)

Thanks, all. Joseph A. Spadaro (talk) 19:14, 7 August 2014 (UTC)

Here's a helpful graphic I ran across from our article on Surveying: "How a surveyor used both traverse and offset measurements to record the location of a shoreline (blue). The circles are rods driven into the ground to create reference points. Dashed lines are traverse lines where the surveyor has recorded whole circle bearings (Azimuths) and distances that allow the reference points' position to be plotted on a map. The red lines are offset measurements." El duderino ^(abides) 05:36, 8 August 2014 (UTC)

Thanks. That's a helpful map. Joseph A. Spadaro (talk) 21:10, 8 August 2014 (UTC)

Primes of the form 7^n+2

Can a number of the form 7ⁿ+2 (for positive integer n) be prime? If not, why not? Many numbers of the general form aⁿ+b are prime, but a quick search using PARI/GP does not find any primes of the form 7ⁿ+2 for small n, so I wonder whether a number of this form can even be prime at all. -- Toshio Yamaguchi 09:29, 3 August 2014 (UTC)

This sounds like a homework question from the first week of a number theory course, so I'll just give you a hint: consider numbers of the form 3x+1... RomanSpa (talk) 12:09, 3 August 2014 (UTC)

Well, it's not a homework question, I am just doing this for my own amusement (which of course you can believe or not). I am not sure I understand the hint you give, but a quick check appears to imply that all of these numbers are divisible by 3. -- Toshio Yamaguchi 12:50, 3 August 2014 (UTC)

Are you familiar with modular arithmetic? Any power of 3x+1 is 1 mod 3. And 7 is of the form 3x+1. Staecker (talk) 13:43, 3 August 2014 (UTC)

I often use PARI/GP to quickly test some numbers for small factors to look for a pattern. This easily adaptable example tests for the first 25 primes as factors in your sequence:

m=prod(i=1,25,prime(i)); for(n=1, 20, p=7^n+2; print(n" "factor(gcd(p,m))[,1]~))

The answer will often be obvious from such output, although you may need a little knowledge of modular arithmetic if you want a complete proof. PrimeHunter (talk) 15:05, 3 August 2014 (UTC)

@PrimeHunter: Indeed, running that code I can already see two different patterns involving the prime factors 3 and 5 repeating with periods 1 and 4. -- Toshio Yamaguchi 19:53, 3 August 2014 (UTC)

You may enjoy working out what happens with 78557×2ⁿ+1, or maybe you already know. We have an article about it. PrimeHunter (talk) 20:42, 3 August 2014 (UTC)

Yes, those can never be prime for any n. :) -- Toshio Yamaguchi 17:36, 4 August 2014 (UTC)