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 Skip Navigation LinksMath Help > Number Theory > Pythagorean Triples > A^4+B^4=C^2

There is no solution...

to the Diophantine equation A4+B4=C2.

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

Here, A2,B2,C are Primitive Pythagorean Triple (assuming without the loss of generality, B2 is even), so coprime integers a,b exist such that

A2=a2-b2
B2=2ab
C=a2+b2

Rearrange the first equation to get A2+b2=a2, so A,b,a is a Pythagorean Triple with b even.  Thus coprime integers c,d exist such that

A=c2-d2
b=2cd
a=c2+d2

Substitute the expressions for b and a back to B2=2ab, so we have

B2=4cd(c2+d2)

As c, d, c2+d2 pairwise coprime, each of them must be a perfect square.  Therefore,

c=e2, d=f2 and c2+d2=g2

Combine these expression to give

e4+f4=g2

with g ≤ g2 = a ≤ a2 < 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.