Goldbach's Conjecture and Zeckendorf's Theorem

Watching the movie A Brilliant Young Mind a couple of nights ago, I heard mention made of Goldbach's Conjecture and thought I should find out about it. This is the introduction to what Wikipedia had to say about it: 

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. It states:

Every even integer greater than 2 can be expressed as the sum of two primes.

The conjecture has been shown to hold up through 4 × 10^18, but remains unproven despite considerable effort.

The expression is not unique as the example of 100 shows:

100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53

I came across Zeckendorf's Theorem when examining the number 24331. It's similar to Goldbach's Conjecture in that it deals with the decomposition of numbers but into Fibonacci numbers, not primes. It states, to quote from Wikipedia again, that:

Every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers.

Now the first few Fibonacci numbers are:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368

The Wikipedia article goes on to state that the theorem has two parts:

Existence: every positive integer n has a Zeckendorf representation.
Uniqueness: no positive integer n has two different Zeckendorf representations.

The example is given of the number 100:

100 = 89 + 8 + 3.

There are other ways of representing 100 as the sum of Fibonacci numbers:

100 = 89 + 8 + 2 + 1

100 = 55 + 34 + 8 + 3

but these are not Zeckendorf representations because 1 and 2 are consecutive Fibonacci numbers, as are 34 and 55.

Sums of Squares

I was prompted to investigate this one when I was checking the OEIS for the prime number 24329 that arose from my daily count of the number of days that I've been breathing in Earth's air. One entry reported that 24329 is a member of sequence A100559 that lists the smallest prime equal to the sum of n distinct squares. 24329 appeared in the case of n=41 so this prime was the smallest one that was equal to the sum of 41 distinct squares. The members of the sequence, up to 24329, are:

5, 29, 71, 79, 131, 179, 269, 349, 457, 569, 719, 971, 1171, 1327, 1601, 1913, 2269, 2593, 2999, 3539, 4099, 4549, 5231, 5717, 6529, 7297, 7879, 8779, 9791, 10711, 11867, 12809, 14081, 15269, 16561, 17863, 19463, 20771, 22541, 24329

The sequence begins with 5 for the case of n=2 where \(1^2+2^2=5\). The examples shown in the OEIS entry are as follows:

\(a(3)=29 \text{ because } 29=2^2+3^2+4^2\)
\(a(4) = 71 = 1^2+3^2+5^2+6^2\)
\(a(5)=79 \text{ because } 79=1^2+2^2+3^2+4^2+7^2\)

It can be seen that the squared numbers do not necessarily need to include 1 nor all of the numbers between 1 and n. It can even include numbers greater than n. For example, 29 (when n=3) does not include 1; 71 (when n=4) begins with 1 but does not include 4 but includes 5 and 6; 79 (when n=5) includes 1, 2, 3, 4 and 7 (but not 5 or 6). 

Interestingly, if the squares of the first n numbers are added together, they do not add to a prime number for the cases n=2 to 42. It may be a mathematical fact that the sum of the squares of the integers in sequence can never add to a prime number. In the list below, I've worked out these sums in Excel and put them in the central column with the corresponding primes from OEIS A100559 in the right-hand column:

1	1
2	5	5
3	14	29
4	30	71
5	55	79
6	91	131
7	140	179
8	204	269
9	285	349
10	385	457
11	506	569
12	650	719
13	819	971
14	1015	1171
15	1240	1327
16	1496	1601
17	1785	1913
18	2109	2269
19	2470	2593
20	2870	2999
21	3311	3539
22	3795	4099
23	4324	4549
24	4900	5231
25	5525	5717
26	6201	6529
27	6930	7297
28	7714	7879
29	8555	8779
30	9455	9791
31	10416	10711
32	11440	11867
33	12529	12809
34	13685	14081
35	14910	15269
36	16206	16561
37	17575	17863
38	19019	19463
39	20540	20771
40	22140	22541
41	23821	24329
42	25585	25913

It can be seen that the primes are always larger than the consecutive sums of squares to the left but not generally larger than the next sum (except in the case n=4 where 71 is larger than 55, the sum of the first 5 squares).

Interprimes

There's always more to discover about prime numbers and numbers that are not themselves prime but are associated with them. One such set of numbers is comprised of so-called interprimes. An interprime is defined by WolframAlpha as follows:

An interprime is the average of consecutive (but not necessarily twin) odd primes. The first few terms are 4, 6, 9, 12, 15, 18, 21, 26, 30, 34, ... (OEIS A024675). The first few even interprimes are 4, 6, 12, 18, 26, 30, 34, 42, 50, 56, 60, ... (OEIS A072568), and the first few odd ones are 9, 15, 21, 39, 45, 69, 81, 93, 99, ... (OEIS A072569).

As well as the odd and even interprimes, there are other subsets as well including the one I discovered yesterday when investigating 24323. It turns out this numbers belongs to OEIS A075288: interprimes which are of the form s*prime, s=13 e.g. 1313 is an interprime and 1313/13 = 101 is prime. The sequence runs as follows:

26, 39, 1313, 4771, 7033, 9607, 11947, 12233, 14963, 15613, 18707, 20527, 24323

The OEIS lists sequences from s = 2 up to s = 21 (A075277-A075296). So yesterday I was halfway between two prime days (24317 and 24329). So I now have a new term and a new concept in my armoury. Of interest in this regard is the following (from WolframAlpha) - doubleclick to enlarge: