Pythagorean Triples and Fermat's Last Theorem

 

The Pythagorean Theorem says that the sum of the squares of the sides of a right triangle equals the square of the hypotenuse. In symbols,

 

a 2 + b 2 = c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaaikdaaaGccqGHRaWkcaWGIbWaaWbaaSqabeaacaaIYaaa aOGaeyypa0Jaam4yamaaCaaaleqabaGaaGOmaaaaaaa@3D57@ .

 

 

Here are a few examples which satisfy the Pythagorean Theorem:

3 2 + 4 2 = 5 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiodadaahaa WcbeqaaiaaikdaaaGccqGHRaWkcaaI0aWaaWbaaSqabeaacaaIYaaa aOGaeyypa0JaaGynamaaCaaaleqabaGaaGOmaaaaaaa@3CDC@ ; 7 2 + 24 2 = 25 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiEdadaahaa WcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaaGinamaaCaaaleqabaGa aGOmaaaakiabg2da9iaaikdacaaI1aWaaWbaaSqabeaacaaIYaaaaa aa@3E58@ ; 8 2 + 15 2 = 17 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIdadaahaa WcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaGynamaaCaaaleqabaGa aGOmaaaakiabg2da9iaaigdacaaI3aWaaWbaaSqabeaacaaIYaaaaa aa@3E5A@ ; 65 2 + 72 2 = 97 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiAdacaaI1a WaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG4naiaaikdadaahaaWc beqaaiaaikdaaaGccqGH9aqpcaaI5aGaaG4namaaCaaaleqabaGaaG Omaaaaaaa@3F22@ .

Are there infinitely many Pythagorean triples? The answer is "YES" for a trivial reason since, for example, for any integer d,

(3d) 2 + (4d) 2 = (5d) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaaIZa GaamizaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaGGOaGa aGinaiaadsgacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0Jaai ikaiaaiwdacaWGKbGaaiykamaaCaaaleqabaGaaGOmaaaaaaa@43A2@

These new triples are not interesting, so we concentrate only on triples with no common factors. Such triples are called primitive Pythagorean triples.

Here are some other triples:

 

(20; 21; 29); (12; 35; 37); (9; 40; 41); (16; 63; 65); (28; 45; 53).

 

 

Pythagorean Triples Theorem:

You will get every primitive Pythagorean triple (a; b; c) with a odd and b even by using the formulas:

a=st MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggacqGH9a qpcaWGZbGaamiDaaaa@39C8@ b= s 2 t 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkgacqGH9a qpdaWcaaqaaiaadohadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG 0bWaaWraaSqabeaacaaIYaaaaaGcbaGaaGOmaaaaaaa@3D69@ , c= s 2 +t 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacqGH9a qpdaWcaaqaaiaadohadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG 0bWaaWraaSqabeaacaaIYaaaaaGcbaGaaGOmaaaaaaa@3D5F@

where s > t ≥1 are chosen to be any odd integers with no common factors.

 

 

Alternatively, since b is even, we could have started with b 2 = c 2 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkgadaahaa WcbeqaaiaaikdaaaGccqGH9aqpcaWGJbWaaWbaaSqabeaacaaIYaaa aOGaeyOeI0IaamyyamaaCaaaleqabaGaaGOmaaaaaaa@3D62@ . Hence ( b 2 ) 2 = ca 2 . c+a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaWaaS aaaeaacaWGIbaabaGaaGOmaaaaaiaawIcacaGLPaaadaahaaWcbeqa aiaaikdaaaGccqGH9aqpdaWcaaqaaiaadogacqGHsislcaWGHbaaba GaaGOmaaaacaGGUaWaaSaaaeaacaWGJbGaey4kaSIaamyyaaqaaiaa ikdaaaaaaa@42D5@  and, we conclude that c+a 2 = u 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam 4yaiabgUcaRiaadggaaeaacaaIYaaaaiabg2da9iaadwhadaahaaWc beqaaiaaikdaaaaaaa@3C50@  and ca 2 = v 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam 4yaiabgkHiTiaadggaaeaacaaIYaaaaiabg2da9iaadAhadaahaaWc beqaaiaaikdaaaaaaa@3C5C@  so that c= u 2 + v 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacqGH9a qpcaWG1bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamODamaaCaaa leqabaGaaGOmaaaaaaa@3C8C@ a= u 2 v 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggacqGH9a qpcaWG1bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamODamaaCaaa leqabaGaaGOmaaaaaaa@3C95@  and b = 2uv where u > v ≥ 1, (u,v) = 1, and u and v have opposite parity.

 

Since the equation a 2 + b 2 = c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaaikdaaaGccqGHRaWkcaWGIbWaaWbaaSqabeaacaaIYaaa aOGaeyypa0Jaam4yamaaCaaaleqabaGaaGOmaaaaaaa@3D57@  has infinitely many solutions, it is natural

to investigate the situation where the exponent 2 is replaced by 3, and then 4, and so on.

 

For example, do the equations

a 3 + b 3 = c 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaaiodaaaGccqGHRaWkcaWGIbWaaWbaaSqabeaacaaIZaaa aOGaeyypa0Jaam4yamaaCaaaleqabaGaaG4maaaaaaa@3D5A@ , a 4 + b 4 = c 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaaisdaaaGccqGHRaWkcaWGIbWaaWbaaSqabeaacaaI0aaa aOGaeyypa0Jaam4yamaaCaaaleqabaGaaGinaaaaaaa@3D5D@ , and a 5 + b 5 = c 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaaiwdaaaGccqGHRaWkcaWGIbWaaWbaaSqabeaacaaI1aaa aOGaeyypa0Jaam4yamaaCaaaleqabaGaaGynaaaaaaa@3D60@

 

have solutions in nonzero integers a, b, c?

 

In 1637 Pierre de Fermat(1601-1665) showed that there is no solution for exponent 4.

 

 

 

 

 

Theorem: Fermat(1637)

 

a 4 + b 4 = c 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaaisdaaaGccqGHRaWkcaWGIbWaaWbaaSqabeaacaaI0aaa aOGaeyypa0Jaam4yamaaCaaaleqabaGaaGinaaaaaaa@3D5D@  has no nontrivial solutions in integers.

 

Proof:

Assume that a 4 + b 4 = c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaaisdaaaGccqGHRaWkcaWGIbWaaWbaaSqabeaacaaI0aaa aOGaeyypa0Jaam4yamaaCaaaleqabaGaaGOmaaaaaaa@3D5B@  where (a, b, c) have no common factor and say b is even. Then,

b 2 =2uv MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkgadaahaa WcbeqaaiaaikdaaaGccqGH9aqpcaaIYaGaamyDaiaadAhaaaa@3B7C@

a 2 = u 2 v 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaaikdaaaGccqGH9aqpcaWG1bWaaWbaaSqabeaacaaIYaaa aOGaeyOeI0IaamODamaaCaaaleqabaGaaGOmaaaaaaa@3D88@

c= u 2 + v 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacqGH9a qpcaWG1bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamODamaaCaaa leqabaGaaGOmaaaaaaa@3C8C@

 

where u > v ≥1, (u, v) = 1; and u and v of opposite parity. In fact, v must be even for if not and u is even, then from a 2 = u 2 v 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaaikdaaaGccqGH9aqpcaWG1bWaaWbaaSqabeaacaaIYaaa aOGaeyOeI0IaamODamaaCaaaleqabaGaaGOmaaaaaaa@3D88@ , we have a 2 13 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaaikdaaaGccqGHHjIUcqGHsislcaaIXaGaeyyyIORaaG4m aaaa@3DBB@  (mod4) which is not possible. Since a 2 + v 2 = u 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaaikdaaaGccqGHRaWkcaWG2bWaaWbaaSqabeaacaaIYaaa aOGaeyypa0JaamyDamaaCaaaleqabaGaaGOmaaaaaaa@3D7D@ , then as above

v=2pq MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAhacqGH9a qpcaaIYaGaamiCaiaadghaaaa@3A93@

a= p 2 q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggacqGH9a qpcaWGWbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyCamaaCaaa leqabaGaaGOmaaaaaaa@3C8B@

u= p 2 + q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacqGH9a qpcaWGWbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyCamaaCaaa leqabaGaaGOmaaaaaaa@3C94@

 

Therefore b 2 =2uv=4pq( p 2 + q 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkgadaahaa WcbeqaaiaaikdaaaGccqGH9aqpcaaIYaGaamyDaiaadAhacqGH9aqp caaI0aGaamiCaiaadghacaGGOaGaaiiCamaaCaaaleqabaGaaGOmaa aakiabgUcaRiaadghadaahaaWcbeqaaiaaikdaaaGccaGGPaaaaa@4536@  and hence p, q, pq and p 2 + q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchadaahaa WcbeqaaiaaikdaaaGccqGHRaWkcaWGXbWaaWbaaSqabeaacaaIYaaa aaaa@3A94@  are squares since p, q, pq and p 2 + q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchadaahaa WcbeqaaiaaikdaaaGccqGHRaWkcaWGXbWaaWbaaSqabeaacaaIYaaa aaaa@3A94@  have no factors in common. Let p 2 = A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchadaahaa WcbeqaaiaaikdaaaGccqGH9aqpcaWGbbWaaWbaaSqabeaacaaIYaaa aaaa@3A88@  and q= B 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadghacqGH9a qpcaWGcbWaaWbaaSqabeaacaaIYaaaaaaa@3997@ . Then A 4 + B 4 = p 2 + q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaahaa WcbeqaaiaaisdaaaGccqGHRaWkcaWGcbWaaWbaaSqabeaacaaI0aaa aOGaeyypa0JaamiCamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadg hadaahaaWcbeqaaiaaikdaaaaaaa@3FF3@  is a square and

 

A 4 + B 4 = p 2 + q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaahaa WcbeqaaiaaisdaaaGccqGHRaWkcaWGcbWaaWbaaSqabeaacaaI0aaa aOGaeyypa0JaamiCamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadg hadaahaaWcbeqaaiaaikdaaaaaaa@3FF3@  = u < u 2 + v 2 =c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaahaa WcbeqaaiaaikdaaaGccqGHRaWkcaWG2bWaaWbaaSqabeaacaaIYaaa aOGaeyypa0Jaam4yaaaa@3C96@  < c 2 = a 4 + b 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaahaa WcbeqaaiaaikdaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaaI0aaa aOGaey4kaSIaamOyamaaCaaaleqabaGaaGinaaaaaaa@3D5B@

 

This sets up an infinite descent chain of squares of whole numbers of the form x 4 + y 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaahaa WcbeqaaiaaisdaaaGccqGHRaWkcaWG5bWaaWbaaSqabeaacaaI0aaa aaaa@3AA8@  which is clearly impossible. (The above argument is called the method of infinite descent and was invented by Fermat.)

 

 

 

To learn more:

 

·       Pythagorean Triples and Fermat's Last Theorem

            Donald Rideout(drideout@mun.ca), Memorial University of Newfoundland