**There is no solution...**

to the Diophantine equation A^{4}+B^{4}=C^{2}.

Note that this is the first stage of proving Fermat's Last theorem for n=4.

Here, A^{2},B^{2},C are Primitive Pythagorean Triple
(assuming without the loss of generality, B^{2} is even), so coprime
integers a,b exist such that

A^{2}=a^{2}-b^{2}

B^{2}=2ab

C=a^{2}+b^{2}

Rearrange the first equation to get A^{2}+b^{2}=a^{2},
so A,b,a is a Pythagorean Triple with b even. Thus coprime integers c,d
exist such that

A=c^{2}-d^{2}

b=2cd

a=c^{2}+d^{2}

Substitute the expressions for b and a back to B^{2}=2ab, so we have

B^{2}=4cd(c^{2}+d^{2})

As c, d, c^{2}+d^{2} pairwise coprime, each of them must be a
perfect square. Therefore,

c=e^{2}, d=f^{2} and c^{2}+d^{2}=g^{2}

Combine these expression to give

e^{4}+f^{4}=g^{2}

with g ≤ g^{2} = a ≤ a^{2} < C

Therefore, assuming C is minimally chosen on the outset, we found a smaller
value for C. Thus, an infinite descent is produced - no solution as
required.

Finally, can you see how to use this information to prove no solution for
Fermat's equation with n=4?

### Related Pages in this website

Infinite Descent Proofs

Pythagorean Triples

Triangles

Formulas for Primitive Pythagorean Triples and
their deriviation -- a way to generate all the triples such that a^2 + b^2
= c^2

Prove that the area of a right
triangle with integer sides is not a perfect square. The proof is
here (with some help from someone from the nrich website)

Puzzle question -- if
m is the product of n distinct primes, how many different right triangles with
integer sides have a leg of length m?

Arithmetic Sequence of Perfect
Squares, page 3 -- If a^2, b^2, c^2 are in arithmetic sequence, why is
their constant difference a multiple of 24? Look at the second answer to
this question for the relationship between Pythagorean Triples and this
arithmetic sequence of squares.

Theorems Involving Perfect Squares
-- answers questions such as "why is the square root of x irrational unless x
is a perfect square?" and other fundamental questions about perfect squares.

Fermat's Therems.

The webmaster and author of this Math Help site is
Graeme McRae.