20070517, 23:44  #1 
May 2004
New York City
3×17×83 Posts 
A Pattern That Is Not
The Law of Small Numbers strikes again !
I was looking for something new in the primes, and thought I had come across a new pattern. I got myself into a lot of trouble here in the forum when I extrapolated a pattern I thought I had found in the Mersenne primes, so I knew I had to be more careful with this "new" pattern. The sequence of Number of Primes < 10^{n} begins: 4,25,168,1229,9592,78498,664579,5761455,50847534,... But 4 and 25 are perfect squares. There "must" be a pattern. And 168 = 13^2  1 is nearly a square, and the correction factor 1 is. And 1229 = 35^2 + 4 is nearly a square, and 4 is. Maybe there's a first or second order pattern ... Well, 9592 = 98^2  12 already starts the drift away from such a pattern. And 78498 = 280^2 + 98 isn't so close to a square, although 98 is close. Maybe there's a third order pattern ... But 664579 = 815^2 + 354, and 354 = 19^2  7 is also "drifting" from being nearly a square. You can see where this went. In short, by the ninth term (sooner if you disagree with what constitutes being "close" to a square), no obvious square pattern persists. Although there might be a fourth order pattern ... Lesson learned. New mathematical patterns and properties are not so easy to come by. 
20090823, 21:01  #2 
Oct 2007
Manchester, UK
17·79 Posts 
I know this thread is a bit old but it interested me so I'm reviving it.
All odd numbers can be generated if you allow subtraction of squares, for example: 1229 = 615^2  614^2 664579 = 332290^2  332289^2 Additionally, all multiples of 4 can be generated by subtracting odd next door neighbours, for example: 168 = 43^2  41^2 9592 = 2399^2  2397^2 Some numbers can't be made using subtraction or addition with only two squares, such as 6 and 50847534. Other numbers like 78498 can though: 78498 = 63^2 + 273^2 
20090823, 21:31  #3 
Aug 2006
3^{2}×5×7×19 Posts 
Yes. Those numbers that can are of 0 density, though only very slightly  they're more common than primes, for example.

20090823, 21:49  #4 
(loop (#_fork))
Feb 2006
Cambridge, England
14360_{8} Posts 
Ooh, that's a nice result; do you happen to know what the asymptotic form of N(sum of two squares less than X) is?

20090824, 07:54  #5  
Sep 2006
Brussels, Belgium
686_{16} Posts 
Quote:
(2n+1)^{2}(2n1)^{2}=8n Both your examples are multiples of 8. Jacob 

20090824, 09:52  #6  
Nov 2003
16444_{8} Posts 
Quote:
A positive integer is the sum of two squares iff all of its prime factors that are congruent to 3 mod 4 appear to an even power. i.e. If N is divisible by a prime p that is 3 mod 4, then it must be divisible by p^(2k) for some positive integer k. 

20090824, 10:06  #7 
Oct 2007
Manchester, UK
17·79 Posts 
Ah yes, it is multiple of 8 for odd numbers, but I didn't mean odd numbers, I meant next door but one neighbors. Not sure why I said odd, let's put that down to a misfiring neuron.
4 = 4  0 8 = 9  1 12 = 16  4 16 = 25  9 20 = 36  16 (n+1)^2  (n1)^2 = 4n Using only addition of squares, any number that is 3 mod 4 cannot be made. I've attached a list of more that cannot be made, and also an explicit list of all integers less than 100 that cannot be made (there are 57, for less than 10^3 there are 670, 10^4 : 7,251, 10^5 : 75,972, 10^6 : 783,659). 
20090824, 11:18  #8 
"Robert Gerbicz"
Oct 2005
Hungary
2674_{8} Posts 

20090824, 11:40  #9  
Nov 2003
2^{2}×5×373 Posts 
Quote:
Every positive integer is the sum of at most 4 squares. I will assume that what you do mean is: "sum of TWO squares". In mathematics, one needs to be precise. Further: Doesn't anyone read what I write? I completely characterized the integers that are the sum of two squares in my prior post. BTW, the proof is easy. I will offer a hint: The Gaussian integers are a unique factorization domain. [UFD]. 

20090827, 15:23  #10  
May 2004
New York City
3×17×83 Posts 
Quote:
as a good teacher would. But can you tell us if this sheds any light on the original post on a supposed pattern among the primes? 

20090827, 17:03  #11  
Aug 2006
3^{2}·5·7·19 Posts 
Quote:
So the numbers seem to act pretty much like you'd expect random numbers to act. If someone wants to fix my expected count for the secondorder sequence and calculate one for the thirdorder, be my guest; it's a little hairy for me to do at the moment. But I'd be surprised if there was a significant deviation from the expected value. 

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