In the Elliptic Realm
Michael Somos
21 Oct 2001
The trigonometric functions have a natural generalization which is
discovered using only the simplest tools possible. These new functions
are part of what can be called "the Elliptic Realm" since they come
from elliptic functions. We construct sequences of values of circular
and hyperbolic trigonometric functions and then generalize them to the
elliptic sequences without requiring any knowlege of elliptic functions.
Section 1. Functions, Tables, and Sequence Construction.
A function will be specified by giving a table of values at equally
spaced points usually starting at zero. The cube function is given by
n : 0 1 2 3 4 5 ...
n*n*n: 0 1 8 27 64 125 ...
which tabulates the function starting at zero. The table of sequence
values can be formed in more than one way using arithmetic operations.
We will be generalizing trigonometric functions so we construct
trigonometric sequence tables to illustrate our construction methods
and also to test potential constructions of generalized sequences.
One good example of a hyperbolic sequence is the following table
n : 0 1 2 3 4 5 6 7 8 9 10 ...
s(n): 0 1 3 8 21 55 144 377 987 2584 6765 ...
constructed starting with initial terms s(0) = 0 and s(1) = 1 .
Each term is three times the previous term minus the term before that.
The recursion equation is s(n) = 3*s(n-1) - s(n-2) for all n.
For example, n = 5 , gives s(5) = 55 = 3*21 - 8 . Note the identity
s(n) = F(2*n) where F() denotes the Fibonacci sequence. We will
call this sequence "Fib2" for future reference. Any number could have
been used instead of s(2) = 3 to construct a sequence. If the number
is greater than 2 , the sequence is hyperbolic, but if less than 2 in
absolute value then the sequence is circular. The linear case is if
the number is 2 and hence s(n) = n by induction.
The trigonometric sequences can be constructed in other ways. For
Fib2 we have, s(4)*s(1) = s(3)*s(2) - s(2)*s(1) = 8*3 - 3*1 = 21 .
We solve this equation to get s(4) = 21 . Similarly, we have also
s(5)*s(1) = s(3)*s(3) - s(2)*s(2) = 8*8 - 3*3 = 55 which we solve
to get s(5) = 55 . These equations and related equations depend on
triples of pairs of positive integers which form a pair of patterns
[4,1] [3,2] [2,1] and [3,1] [2,2] [1,1]
[6,1] [4,3] [3,2] and [5,1] [3,3] [2,2]
... ... ... ... ... ...
[2n+2,1] [n+2,n+1] [n+1,n] and [2n+1,1] [n+1,n+1] [n,n]
In other words, the following equations
s(2*n+2)*s(1) = s(n+2)*s(n+1) - s(n+1)*s(n) , and
s(2*n+1)*s(1) = s(n+1)*s(n+1) - s(n)*s(n) ,
hold for all positive n . These equations can be solved for s(n)
where n is greater than 2 given any values for s(2) and s(1) but s(1)
must be non-zero. When s(1) = 1 and s(2) = x , these sequences are
named Chebyshev polynomials of the second kind. Now we seek a simple
generalization of trigonometric sequences.
Section 2. Sums of Four Squares and Elliptic Sequences.
To discover generalized sequences we need a starting point. Among
many possible starting points, I have come across what I think is the
easiest so far. It begins with the sum of four non-zero squares. It
is known that every positive number is the sum of four squares, but
not always four non-zero squares. The smallest such number is 4 since
4 = 1+1+1+1 is a sum of four non-zero squares. Next on the list is
7 = 4+1+1+1 and then 10 = 4+4+1+1 .
Suppose we now require the number to be a sum of four non-zero
squares in more than one way. In this case, the smallest such number
is 28 and we have the following three sums
[5,1,1,1] [4,2,2,2] [3,3,3,1]
28 = 25+1+1+1 = 16+4+4+4 = 9+9+9+1 .
Above each square appears its square root. The three square root
quadruples have many arithmetic properties. For example, if we
take the product of the four numbers of each quadruple we get
[5,1,1,1] [4,2,2,2] [3,3,3,1]
5 = 5*1*1*1 32 = 4*2*2*2 27 = 3*3*3*1
Note that 5 = 32 - 27 . Now we use each number of each quadruple
to index into the Fib2 sequence mentioned earlier. Recall that
s(2) = 3 , s(3) = 8 , s(4) = 21 , s(5) = 55 , yielding
[5,1,1,1] [4,2,2,2] [3,3,3,1]
55 = 55*1*1*1 567 = 21*3*3*3 512 = 8*8*8*1
as the calculated product of corresponding terms and 55 = 567 - 512 .
These quadruples have unexpected arithmetic properties when indexed
into trigonometric sequences. Let's explore these properties further.
The next smallest number which is the sum of four non-zero squares
in three ways is 42. We have the following sums
[6,2,1,1] [5,3,2,2] [4,4,3,1]
42 = 36+4+1+1 = 25+9+4+4 = 16+16+9+1
and again the square root quadruples appear above. Note again that
we get the equality of difference of products as given by
[6,2,1,1] [5,3,2,2] [4,4,3,1]
12 = 6*2*1*1 60 = 5*3*2*2 48 = 4*4*3*1
and 12 = 60 - 48 . Also, taking the product of corresponding indexed
terms of the Fib2 sequence gives the expected result
[6,2,1,1] [5,3,2,2] [4,4,3,1]
432 = 144*3*1*1 3960 = 55*8*3*3 3528 = 21*21*8*1
which is 432 = 3960 - 3528 . Let's summarize our findings. We have
found some triples of quadruples of positive integers with the
following properties :
1) (Lagrange) The sum of squares of each quadruple is equal to
the sum of squares of the other two quadruples.
2) The product of the first quadruple is the product of the
second quadruple minus the product of the third quadruple.
3) (Weierstrass) Same as property 2 except we use the product of
the corresponding indexed terms of trigonometric sequences.
We will call these "Pfaffian triples" since they come from Pfaffians.
The Weierstrass property relates Pfaffian triples to numeric sequences.
We will call non-trigonometric sequences that satisfy the Weierstrass
property "elliptic sequences".
A reasonable goal is now to find all Pfaffian triples and all the
sequences of numbers satisfying the Weierstrass property. The next sum
of four non-zero squares in more than two ways is 52, where
[7,1,1,1] [5,5,1,1] [5,3,3,3] [4,4,4,2]
52 = 49+1+1+1 = 25+25+1+1 = 25+9+9+9 = 16+16+16+4
We seek to select a Pfaffian triple of quadruples. In this case it is
[7,1,1,1] [5,3,3,3] [4,4,4,2]
7 = 7*1*1*1 135 = 5*3*3*3 128 = 4*4*4*2
and 7 = 135 - 128 . Can we generalize this? There are a few clues
here. For example, the first and third Pfaffian triples compared
[5,1,1,1] [4,2,2,2] [3,3,3,1]
[7,1,1,1] [5,3,3,3] [4,4,4,2]
might suggest there is a simple arithmetic progression here. Does
[9,1,1,1] [6,4,4,4] [5,5,5,3]
give us another? Yes, it does, and the general pattern is given by
[2*n+1,1,1,1] [n+2,n,n,n] [n+1,n+1,n+1,n-1]
This gives us one infinite set of triples. Are there any others? Yes.
For example, starting from the second triple we get a pattern
[6,2,1,1] [5,3,2,2] [4,4,3,1]
[8,2,1,1] [6,4,3,3] [5,5,4,2]
[10,2,1,1] [7,5,4,4] [6,6,5,3]
... ... ...
[2*n+2,2,1,1] [n+3,n+1,n,n] [n+2,n+2,n+1,n-1]
which is similar to the first pattern. From just these two patterns
we can construct sequences in a very simple way.
Section 3. Constructing Elliptic Sequences.
Now that we have infinite sets of Pfaffian triples we can use
them to construct elliptic sequences {a(n)}. We start with the values
of a(1),a(2),a(3),a(4). The first triple equation for a sequence is
a(5)*a(1)*a(1)*a(1) = a(4)*a(2)*a(2)*a(2) - a(3)*a(3)*a(3)*a(1)
so if we know the value of a(1),a(2),a(3),a(4), then the equation is
linear in a(5) and we solve for it (assuming non-zero a(1)). Now,
given a(5), the next equation is linear in a(6)
a(6)*a(2)*a(1)*a(1) = a(5)*a(3)*a(2)*a(2) - a(4)*a(4)*a(3)*a(1)
and we solve for it (assuming non-zero a(1) and a(2)). So, given the
equations from the two patterns we can solve for all the rest of the
terms of the sequence. This gives a construction of new sequences,
but what about other patterns? For example, starting from the first
and the second Pfaffian triple leads to the pattern
[5,1,1,1] [4,2,2,2] [3,3,3,1]
[6,2,1,1] [5,3,2,2] [4,4,3,1]
[7,3,1,1] [6,4,2,2] [5,5,3,1]
... ... ...
[n,n-4,1,1] [n-1,n-3,2,2] [n-2,n-2,3,1]
Do the sequences fit into the other patterns? Yes, but the proof
is not obvious and requires work. What do non-trivial examples of
elliptic sequences look like? A simple example is
n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
a(n): 0 1 1 1 -1 -2 -3 -1 7 11 20 -19 -87 -191 -197 ...
which satisfies a(n)*a(n-4) = a(n-1)*a(n-3) - a(n-2)*a(n-2) and
many other equations. A closely similar example is
n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
a(n): 0 1 1 -1 1 2 -1 -3 -5 7 -4 -23 29 59 129 ...
which satisfies a(n)*a(n-4) = a(n-1)*a(n-3) + a(n-2)*a(n-2) and
other equations. In fact, the odd indexed terms alternate in sign,
and with the sign removed the resulting positive sequence
n: 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
a(n): 1 1 1 1 2 3 7 23 59 314 1529 8209 83313 ...
satisfies the same recursion equation. It is called the Somos-4
sequence -- the simplest member of a large family of sequences.
All this and more is part of the Elliptic Realm. The elliptic
sequences are simply related to Jacobi theta functions and Weierstrass
sigma functions evaluated at muliples of a number. Some special cases
were called "elliptic divisibility sequences" by Morgan Ward in 1948.
I have demonstrated that they can easily be discovered and constructed
using only simple tools, and without knowledge of elliptic functions.
One starting point is Pfaffian triples and the Weierstrass property.