# The Congruent Number 5 Problem

Michael Somos described this problem to me; it is a problem he worked on early in his career.

Find a two rational numbers a and b such that a^{2}-5b^{2}
and a^{2}+5b^{2} are both squares of rational numbers.

Any solution can be converted to an integer solution by multiplying all numbers by the LCM of the denominators of a and b.

History

This problem has quite a history going back to classical Greece; it is
a specific class of solutions to Diophantine equations involving
squares in arithmetic progression. I'll note that for integer a and
k, if a^{2}+k and a^{2}-k are both squares, then the valid
values k are: 24, 96, 120, 216, 240, 336, 384, ... (OEIS sequence
A256418).

Fibonacci wrote a not-very-well-known book The Book of Squares addressing this more general problem, which is called the congruum problem. Example solutions include:

29^{2} - 840 = 1 = 1^{2} ; 29^{2} + 840 = 1681 = 41^{2}

37^{2} - 840 = 529 = 23^{2} ; 37^{2} + 840 = 2209 = 47^{2}

Sequences with the Generating Function p(t)/(1-ct+t^{2})

The even-indexed Fibonacci numbers F_{2n} = 0, 1, 3, 8, 21, 55,
144, 377, 987, 2584, 6765, ... (Sloane's A1906) have generating
function

g.f.: *t/(1-3t+t^{2})

and so are defined by the recurrence

A_{0} = 0; A_{1} = 1; A_{n} = 3A_{n-1} - A_{n-2}

The Chebyshev_polynomials have the generating functions and recurrence definitions

g.f.: 1-*xt/(1-2xt+t^{2})

T_{0} = 1; T_{1} = x; T_{n} = 2xT_{n-1} - T_{n-2}

and

g.f.: 1/(1-2*xt+t^{2})

U_{0} = 1; U_{1} = 2x; U_{n} = 2xU_{n-1} - U_{n-2}

Michael Somos, "In the Elliptic Realm", writes:

[...] the following equations

s(2n+2)*s(1) = s(n+2)*s(n+1) - s(n+1)*s(n) , and

s(2n+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
related to Chebyshev polynomials of the second kind as follows

s(n) = U(n-1, x/2).

Since "s(1)" is 1, the definitions are equivalent to:

s(1) = 1

s(2) = x

s(2n+1) = s(n+1)^{2} - s(n)^{2}

s(2n+2) = s(n+2)s(n+1) - s(n+1)s(n)

Letting n=1, we get:

s(3) = x^{2}-1

s(4) = s(3)x - x = x^{3} - 2x

Letting n=2, we get:

s(5) = s(3)^{2} - s(2)^{2} = x^{4}-2x^{2}+1 - x^{2}
= x^{4} - 3x^{2} + 1

s(6) = s(4)s(3) - s(3)s(2) = (x^{3}-2x)(x^{2}-1) - (x^{2}-1)x
= x^{5} - 4x^{3} + 3x

As Somos writes, these polynomials can be converted to the
Chebyshev polynomials of the second kind by
replacing x with 2x everywhere (so for example 3x^{2} becomes
12x^{2}) and offsetting the index by one (so for example s(5)
becomes U(4)).

Congrua of the Form kb^{2} for Small Integer k

Fermat took up this problem and looked for values of the congruum that
are equal to some small integer times a square. He showed that there
are no solutions of the form a^{2}±b^{2}, a^{2}±2b^{2},
or a^{2}±3b^{2}; something of the form a^{2}±4b^{2} is
ruled out by the a^{2}±b^{2} result; this left
a^{2}±5b^{2} as the first open problem.

Somos gives the following four sequences, which jointly are generated
by a recurrence, and give solutions that sometimes involve a minus
sign (and thus an i in the number which, when squared, is one of the
values a^{2}±5b^{2}):

b = W(n) = 0, 1, -12, -2257, 1494696, 8914433905, -178761481355556, -62419747600438859233, ... (A129206)

X(n) = 1, 1, -49, -4799, 4728001, 18618840001, -767067390499249, -54213419267800732799, ... (A129207)

a = Y(n) = 1, 2, 41, 1562, 3344161, -7118599318, 654686219104361, -128615821825334210638, ... (A129208)

Z(n) = 1, 3, 31, 5283, -113279, 21166249443, -518493692732129, 189797666150873887683, ... (A129209)

The first few examples are:

2^{2}-5×1^{2} = -1 = i^{2} ; 2^{2}+5×1^{2} = 9 = 3^{2}

41^{2}-5×12^{2} = 961 = 31^{2} ; 41^{2}+5×12^{2} = 2401 = 49^{2}

1562^{2}-5×2257^{2} = -23030401 = 4799i^{2} ;
1562^{2}+5×2257^{2} = 27910089 = 5283^{2}

References

[1] Michael Somos, Step into the Elliptic Realm, 2000 Mar 23.

[2] Michael Somos, In the Elliptic Realm, 2015 Feb 12.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2018 Aug 27. s.11