The first pattern is x4+y4 = z2, and if this has no solution, then neither does x4+y4 = z4. In other words, we are proving Fermat's last theorem for n = 4. Two fourth powers cannot sum to a fourth power.
We need to consider all these patterns in one go. This is the nature of the proof. The reasoning goes something like this.
If pattern 1 has a solution then there is a smaller triple that satisfies pattern 2. And if pattern 2 has a solution then there is a smaller triple that satisfies pattern 1. Thus every solution implies a smaller solution of the same type, and since positive integers cannot descend forever, there is no solution.
This approach is called the "method of infinite descent", and it is a consequence of induction. If there is a solution there is a least solution, and yet there is always a smaller one, which is a contradiction.
What makes one triple smaller than another? This can be a bit unclear, especially when the triples fit different patterns. The answer is the value of the right hand side. (If the right side is z2, the value is z2, not z.) If one triple has a lesser "value" than another, then it is smaller.
Whenever the coprime lemma applies, we automatically assume coprime variables. After all, this can only make the right side smaller.
x2 ← u2 - v2
y2 ← 2uv
z ← u2 + v2
Since 2uv is a square, and u and v are coprime, u and v are s2 and 2t2. Therefore x2 becomes s4-4t4 or 4t4-s4. Rewrite this as x2+4t4 = s4 or x2+s4 = 4t4. These are patterns (1) and (2) below.
Each time we defer to another pattern, you should verify that the right side has become smaller. In this case the right side is u2 or v2. Either of these is smaller than z, which is smaller than z2, so we're ok.
(½x)2 + (½y2)2 = (z2)2
Remember that ½x is odd, and so is z2. Characterize the triple as follows.
½x ← u2 - v2
½y2 ← 2uv
z2 ← u2 + v2
With u and v coprime, the second equation shows u and v are squares. The third equation becomes s4+t4 = z2. With z2 < 4z4, we have no trouble deferring to pattern (0).
x ← u2 - v2
2y2 ← 2uv
z2 ← u2 + v2
The second shows u and v are squares, and the third becomes s4+t4 = z2. With z2 < z4, we defer to pattern (0).
Let either x or z be even, whence the other is also even. If y is even, divide x by 4 and y and z by 2 to find a smaller solution. Thus y is odd. This means x = 2 mod 4, else y would be even. Divide through by 4 to get this.
(½x)2 + (y2)2 = (½z2)2
The sum of two odd squares is an even square, and a mod 4 argument shows this is impossible.
Patterns (0), (1), and (2) all fail together, and as a corollary, the sum of fourth powers is never a fourth power. If that's all you needed to know, you can skip the rest of this page, but if you're interested, there are many more patterns with no solutions. Some of these act as lemmas for future theorems.
If y is odd, characterize the triple as y2 = u2 - v2 and z2 = u2 + v2. Multiply these together to show u4 - v4 = (yz)2, pattern (3). Note that y < z, hence y2z2 < z4.
x2 ← u2 - v2
2y2 ← 2uv
z ← u2 + v2
Thus u and v are squares, and x2 = s4-t4, pattern (3).
Let x and z be even. If z is divisible by 4 we can divide through and find a smaller solution, so let z = 2 mod 4. This makes y odd. Divide through by 4 and characterize as follows.
y2 ← u2 - v2
½x2 = 2uv
½z = u2 + v2
This implies y2 = s4-t4, pattern (3).
x2 ← ±(u2 - 2v2)
y ← 2uv
z2 ← u2 + 2v2
Multiply the first and third equations together. Now (xz)2 is the difference between a fourth power and 4 times a fourth power, patterns (1) and (2).
±x2 ← u2 - 2uv - v2
y2 ← u2 + 2uv - v2
z ← u2 + v2
Multiply the first two equations together and the left side becomes ±(xy)2. Add 8u2v2 two both sides and the right side becomes (u2+v2)2. This produces the following two equations.
(xy)2 + 8u2v2 = (u2+v2)2
(xy)2 + (u2+v2)2 = 8u2v2
The second equation is impossible mod 4, so concentrate on the first. Write 2st = 2uv, and s2+2t2 = u2+v2.
In this characterization, v might be 0, but u cannot be. thus we can replace v with st/u.
s2+2t2 = u2 + s2t2/u2
(u2)2 - (s2+2t2)u2 + s2t2 = 0
Apply the quadratic formula. The discriminant is s4+4t4, which must be a square. This never happens, thanks to pattern (4). Of course s or t could be 0; pattern (4) doesn't preclude that. Thus v = 0, and the unique coprime solution emerges: 14 + 14 = 2×12.
after characterizing the triple, we find u2+v2 = 3z2. This is impossible.
If 3 does not divide x and y, there is no solution, thanks to pattern (7), so let 3 divide x and y.
If 3 divides z we can divide through and find a smaller solution, so 3 does not divide z, and x = 3 or 6 mod 9. Let y be even and characterize the triple as follows.
x/3 ← u2 - v2
y2/3 ← 2uv
z2 ← u2 + v2
Remember that u and v are coprime, and look at 6uv = y2. Thus u and v are 6s2 and t2, or 3s2 and 2t2. Substitute in the third equation.
36s4 + t4 = z2
9s4 + 4t4 = z2
Since z2 is less than 9z4, we defer to patterns (10) and (11).
Next let y be odd and find the following characterization.
y2/3 ← u2 - v2
x/3 ← 2uv
z2 ← u2 + v2
Use the last equation to assign s2-t2 and 2st to u and v. Then substitute in the first equation.
±y2/3 = s4 - 6s2t2 + t4
The left side can be viewed as ±3 times a square. Look at this equation mod 8. Exactly one of s and t is even, hence two of the three terms on the right drop to 0. Now a fourth power is ±3 times a square. This is impossible mod 8, so we're done.
Suppose x is divisible by 3. It must be 3 or 6 mod 9, and 3 does not divide y. Derive the following equation, and note that it is impossible mod 3.
(x/3)2 + y4 = (z2/3)2
Thus the terms are pairwise coprime. There are two different characterizations, depending on whether y is even or odd. In each case z2 = u2+v2.
Let y be odd, whence u2-v2 = 3y2. Multiply this by the equation for z2 and find 3(yz)2 = u4-v4. Since u2 < z2, u4 < z4, and we defer to pattern (9).
If y is even then 2uv = 3y2. Thus u and v are 6s2 and t2, or 3s2 and 2t2. Substitute these expressions to set z2 equal to 36s4 + t4 or 4s4 + 9t4. These are patterns (10) and (11) respectively.
Let x be even, and z odd, and refer to the characterization of such triples.
±y2 ← u2 - 3v2
x ← 2uv
z2 ← u2 + 3v2
Multiply the first and third together to show (yz)2 is the difference between a fourth power and 9 times a fourth power. These are patterns (7a) and (8).
Next let z be even, whence x and y are odd. In this case the characterization gives z2/2 = u2+3v2. Since z2/2 is even, u and v have the same parity. This contradicts the characterization, i.e. it makes x and y even as well.
Let 2 divide x and z, while 3 does not. For a minimal solution, z = 2 mod 4, and y is odd. Divide through by 4 and characterize.
3y2 ← u2 - v2
x2/2 = 2uv
z/2 = u2 + v2
The second equation shows u and v are squares, and the first equation sets 3 times a square equal to the difference between fourth powers, which is pattern (9).
Next let 3 divide x and z while 2 does not. For a minimal solution, z is not divisible by 9, and y is not divisible by 3. Divide through by 9 and characterize.
x2/3 ← u2 - v2
2y2 ← 2uv
z/3 ← u2+v2
Once again u and v are squares, and the first equation leads to pattern (9).
Finally let 6 divide x and z, but not y, so that we may have a minimal solution.
y2 ← u2 - v2
x2/6 ← 2uv
z/6 ← u2 + v2
This time u and v are a square and 3 times a square, and the first equation sets x2 to the difference between a fourth power and 9 times a fourth power, patterns (7a) and (8).
Let 2 divide y, while 3 does not divide x. Thus 3y2/2 = 2uv, and x2 = u2-v2. The first equation makes u and v a square and 3 times a square, and the second produces patterns (7a) and (8).
Let 3 divide x, while 2 does not divide y. Thus 2x2/3 = 2uv, and y2 = u2-v2. The first equation makes u and v a square and 3 times a square, and the second produces patterns (7a) and (8).
finally 3 divides x and 2 divides y. Divide through by 36 for a coprime triple. Thus y2/2 = 2uv, and x2/3 = u2-v2. With u and v squares, the second equation leads to pattern (9).
This completes the circle; all patterns up to this point have no solution, save the one solution for pattern (6). But there are more patterns to consider.
If z is even then z2 is divisible by 4, which is impossible, since the characterization of z2/2 is an odd number.
x4/4 + 3y4 = z2/4
Thus z and x are odd. Characterize this triple and write 2y2 = 2uv, and x2 = u2-3v2. Substitute squares into the second equation to get x2 = ±(s4-3t4). This either sets a sum of squares equal to 3 times a square, which is impossible mod 4, or it creates pattern (12).
Divide through by 4, and only the second term remains even. Characterize, and write y2/2 = 2uv, and x2 = u2-3v2. Once again x2 is the difference between a fourth power and 3 times a fourth power. As we saw in the last pattern, this is impossible, or it creates pattern (12).
The following equation is equivalent.
(z+x4) × (z-x4) = 243y8.
If 3 divides both factors it divides x, and we ruled that out. Thus 243 belongs to one of the two factors alone.
Let y be odd, whence one factor is an eighth power, and the other is 243 times an eighth power. Subtract the two equations to get ±2x4 = u8-243v8. Reduce this mod 17. Both u and v cannot drop to 0, as they are coprime. An eighth power is 0 1 or -1. In addition to these, a fourth power could be ±4. If u or v is 0 the right side is ±1 or ±5, while the left side is 0, ±2, or ±8. This cannot be reconciled, so both u and v are nonzero. The right side becomes ±4 or ±6. The left side is still 0, ±2, or ±8. We are forced to conclude that y is even.
±x4 ← u2 - 3v2
9y4 = 2uv
z = u2+3v2
To avoid squares summing to thrice a square, the first equation produces +x4. Thus x4+3v2 = u2.
U and v are fourth powers, with coefficients of 8 and 9. If 3 divides u it divides x, hence 9 belongs to v. That leaves two possibilities.
Set u = s4 and v = 72t4, giving x4 + 15552t8 = s8. Thisis pattern (13).
Setting u = 8s4 and v = 9t4 leads to the following.
x4 + 243t8 = 64s8
This is impossible mod 8, unless x and t are even. With 64 on the right, x must be divisible by 4. This makes s even, and we can divide through for a smaller solution.
16x8 + 243y8 = z2
This is impossible mod 4, hence x and z are odd.
If 3 divides x and z then 27 divides z, and 3 divides y, and 81 divides z. We could divide z by 81 and x and y by 3 to find something smaller. Therefore, all terms are coprime.
±x4 ← u2 - 3v2
36y4 ← 2uv
z ← u2+3v2
This time u and v are fourth powers with coefficients of 2 and 9. If 3 divides u it divides x, hence 9 belongs to v.
Set u = s4 and v = 18t4, giving x4 + 972t8 = s8. This is pattern (13).
Set u = 2s4 and v = 9t4 and derive x4 + 243t8 = 4s8. This is pattern (18).
If x and z are divisible by 3, we can make x divisible by 9 and find a smaller solution, as we have done before. Thus the terms are coprime.
The following equation is equivalent.
(2z4+x2) × (2z4-x2) = 243y8.
With x and z nonzero mod 3, 243 divides the first factor. Othere than this, both factors are eighth powers.
Add the equations together to get this.
4z4 = u8+243v8
With 4z4 < 4z8, this is pattern (15).