The assertion of the Fermat equation seems to be incorrect at a basic level. True? [formatting note: message set for line wraps at 95 characters] Symbols: == congruent, [ ] subscript, | vertical bar used as a tabulation column marker Preconditions For x^3 + y^3 = z^3 working mod 8, identify primitive triples where gcd(x, y, z) = 1 (all other triples are generated by multiplying through by arbitrary d^3). The cubes of even integers are congruent with 0 mod 8, while the cubes of odd integers are congruent with 1, 3, 5, 7 mod 8. The properties of the primitive triple also require that x, y, and z be relatively prime in pairs. If x, y, and z form a primitive triple, then one of these conditions can hold: either one of x and y must be even, with the other odd, y == z; one of x and y is 1 mod 8 with the other 7 mod 8, z even; one of x and y is 3 mod 8 with the other 5 mod 8, z even. The possibility of x and y both being odd fails to yield integer values in the procedure outlined below, leaving odd-even combinations of x and y, x being designated the even element and y the odd element, with the added condition that y > x. Concept Taking advantage of 1 + 3 + 5 + ... + (2n - 1) = n^2 n^3 is n + 3n + ... + (2n - 1)n = n^2n a sum of cubes of x and y would be (x + y) + 3(x + y) + ... + (2x - 1)(x + y) + (2x + 1)y + ... + (2y - 1)y if y > x and x^3 + y^3 = z^3 becomes (x + y) + 3(x + y) + ... + (2x - 1)(x + y) + (2x + 1)y + ... + (2y - 1)y = (2z - 1)z + (2z -3)z + ... + z These series can be condensed with the approximations F[ml], X[ml], and Z[ml], where m is the power being considered and l indicates the level of expansion of the terms. Adopting the convention that the sum of cubes is expanded starting at the disjunction between terms (2x - n)(x + y) and terms (2x + n)y, while the cube of the single integer is expanded starting from the highest-valued term (2z - 1)z F[31] = (2x -1)(x + y) F[32] = (2x -1)(x + y) + (2x + 1)y F[33] = (2x - 3)(x + y) + (2x -1)(x + y) + (2x + 1)y F[ml] = X[ml] (to be explained) Z[31] = (2z - 1)z Z[32] = (2z - 1)z + (2z - 3)z Z[33] = (2z - 1)z + (2z - 3)z + (2z - 5)z The terms of the sum of cubes is expanded alternately between the terms (2x - n)(x + y) and terms (2x + n)y. In general X[3i](x,y) + u[3i] = - (n^2)x - (2n - 1)y + (2n)x^2 + 2(2n - 1)xy + u[3i] (i = 2n - 1) = - (n^2)x + (2n)x^2 + 2(2n)xy + u[3i] Z[3i](z) + v[3i] = - i^2z + 2iz^2 + v[3i] (i = n) where u[3i] and v[3i] are error terms. F[ml] is distinct from X[ml] in the calculation of the error terms u[3i] = (x^3 + y^3) - X[3i](x,y) w[3i] = z^3 - F[3i](x,y) For the seventh level of expansions of the sum of cubes i = 7 = 2(4) -1, n = 4; for the single-integer cube i = 7 = n X[3i](x,y) = - (n^2)x - (2n - 1)y + (2n)x^2 + 2(2n - 1)xy +u[3i] (i = 7 = 2(4) - 1) X[37](x,y) = - (n^2)x - (2n - 1)y + (2n)x^2 + 2(2n - 1)xy +u[3i] = - (4^2)x - (2(4) - 1)y + (2(4))x^2 + 2(2(4) - 1)xy +u[37] = - 16x - 7y + 8x^2 + 14xy +u[37] Z[3i](z) = - i^2z + 2iz^2 + v[3i] (i = 7) Z[37](z) = - 7^2z + 2(7)z^2 + v[37] = - 49z + 14z ^2 + v[37] The sole purpose of the construction work here is to develop alternative expressions for cubes and sums of cubes that can be operated on; however, it should be pointed out that the expansions can be carried out to any level of precision desired, with the expansion of the sum of cubes being adjusted to account for the unequal number terms with factors (x + y) versus those with factors y. The Fermat equation for cubes can now be rewritten x^3 + y^3 = z^3 F[37] + w[37] = Z[37] + v[37] but if the supposition is that x^3 + y^3 = z^3, then it can be supposed that X[37] + u[37] = Z[37] + v[37]. Returning to the seventh level of expansions, rewriting the nonlinear expressions X[37](x,y) = - 16x - 7y + 8x^2 + 14xy + u[37] Z[37](z) = - 49z + 14z^2 + v[37] as the linear expressions X[37](x[1],x[2],x[3],x[4]) = - 16x[1] - 7x[2] + 8x[3] + 14 x[4] + u[37] Z[37](z[1],z[2]) = - 49z[1] + 14z[2] + v[37] the nonlinear values are a subset of the linear values. A failure to find any linear solutions would necessarily also be a failure for the nonlinear system. Methods Rather than compare F[37] + w[37] directly with Z[37] + v[37], the comparison is X[37] + u[37] = 8k[9] + r[8] Z[37] + v[37] = 8k[9] + r[8] where 8k[9] + r[8] is either 8k[9] + r[8] = (8j)^3 + 3(8j)^2r + 3(8j)r^2 + r^3 = 8(8^2j^3 + 8(3)j^2r + 3jr^2) + r^3 = 8k[9] + r[8] = (8j + r)^3 (cube of one integer) 8k[9] + r[8] = 8(8^2i^3 + 8(3)i^2r + 3ir^2 + 8^2j^3 + 8(3)j^2r + 3jr^2) + (r^3 + s^3) = 8k[9] + r[8] = (8i + r)^3 + (8j + s)^3 (sum of cubes) The supposition, once again, is that these renderings are interchangeable. An additional modification is made 8k[9] = 8(8k[10] + r[9]) = 64k[10] + 8r[9] giving X[37] + u[37] = 64k[10] + 8r[9] + r[8] Z[37] + v[37] = 64k[10] + 8r[9] + r[8] for a final working set of equations - r[8] - 16x[1] - 7x[2] + 8x[3] + 14x[4] + u[37] - 8r[9] - 64k[10] = 0 - r[8] - 49z[1] + 14z[2] + v[37] - 8r[9] - 64k[10] = 0 r[8] can be reduced to a constant based on the remainders mod 8 of x, y, and z and the preconditions outlined above. A specific example x[1] = x = 34 = 8(4) + 2 x[2] = y = 43 = 8(5) + 3 gcd(x,y) = 1 x^3 = 34^3 = (8(4) + 2)^3 = 8(4912) + 8 y^3 = 43^3 = (8(5) + 3)^3 = 8(9935) + 27 34^3 + 43^3 = (8(4912) + 8) + (8(9935) + 27) = 8(14847) + 35 = 118811 the preconditions dictate that if y == 3 mod 8, then so is z, whose cube can be represented by 8j + 27, for some appropriate j. That is X[37] + u[37] = 8k[9z] + 27 = 8k[9x] + 35 the sum of cubes can be represented in these two different ways. Concentrating on the sum of cubes, the working equation is solved for the two (supposedly identical) cases - 35 - 16x[1] - 7x[2] + 8x[3] + 14x[4] + u[37] - 8r[9] - 64k[10] = 0 - 27 - 16x[1] - 7x[2] + 8x[3] + 14x[4] + u[37] - 8r[9] - 64k[10] = 0 using a parametric linear solution where the parameters are constrained by the requirements of the variables r[8], x[1], x[2], x[3], x[4] , u[37]. This requires 7 parameters: t[1], t[2], t[3], t[4], t[5], t[6], t[7]. r[8] = t[1] x[1] = x a function of t[1], t[2] x[2] = y a function of t[1], t[2], t[3] x[3] = x[1]^2 a function of t[1], t[2], t[3], t[4] x[4] = x[1] x[2] a function of t[1], t[2], t[3], t[4], t[5] u[37] = (x^3 + y^3) - X[37] a function of t[1], t[2], t[3], t[4], t[5], t[6] r[9], k[10] functions of t[1], t[2], t[3], t[4], t[5], t[6] , t[7] The parameters are set sequentially, starting with r[8] = t[1] and assuring that the remaining parameters are set to yield the appropriate values. If only the first 6 parameters are set (t[7] = 0), the parametric solution for k[10] happens to equal k[9]. The justification for using the linear parametric method is that it is assured that if x[0], y[0] are particular solutions of ax + by = c then ALL solutions are given by x = x[0] + (b/d)t, y = y[0] - (a/d)t The working equation consists of a "binomial part" and a "sum-of-cubes part". The sum-of-cubes part is always held to the value of x^3 + y^3 and is linked to the binomial part by t[1] = r[8]. The derived value of k[9] is entirely dependent on the parametric solution for u[37], set by t[6]. Returning to the example, if t[1] = r[8] is set to 35, the constrained solution for k[9] is 14847; but a value of 14848 results from setting r[8] to 27. The proposed value satisfies the equation 8(14848) + 27 = 118784 + 27 = 118811 = 34^3 + 43^3 but does it conform to the requirements of the cube of a single integer? If it does, then 8(14848) + 27 = (8j + r)^3 = 8(8^2j^3 + 8(3)j^2r + 3jr^2) + 3^3 = 8(8^2j^3 + 8(3)j^2r + 3jr^2) + 27 14848 = 8^2j^3 + 8(3)j^2(3) + 3j3^2 = 8(8j^3 + 9j^2) + 27j = 8k[10] + r[9] = k[9z] One of the preconditions was that y > x. Any proposed j must allow a derived z^3 to at least be greater than or equal to y^3 which, being congruent with the proposed z^3, can be constructed using a value k (in this case 5) that also adheres to the form 8^2j^3 + 8(3)j^23 + 3j3^2 = 64j^3 + 72j^2 + 27j in other words, k = j = 5 forms the base entry in a table of candidate j generating candidate k[9] j^3 | j^2 | j | 64j^3 | 72j^2 | 27j | sum | sum | mod 8 125 | 25 | 5 | 8000 | 1800 | 135 |9935 | 7 216 | 36 | 6 | 13824 | 2592 | 162 |16578 | 2 343 | 49 | 7 | 21952 | 3528 | 189 |25669 | 5 512 | 64 | 8 | 32768 | 4608 | 216 |37592 | 0 729 | 81 | 9 | 46656 | 5832 | 243 |52731 | 3 1000 | 100 | 10 | 64000 | 7200 | 270 |71470 | 6 1331 | 121 | 11 | 85184 | 8712 | 297 |94193 | 1 1728 | 144 | 12 | 110592 | 10368 | 324 |121284 | 4 43^3 = 8(9935) + 27 Testing indicates that each k[9] in the table of candidates is around 1.4 times the preceding entry, converging toward 1 as the values increase. On the other hand, any proposed j can be no more than twice the base value, since the precondition that y is the larger of the pair x, y limits x to at most y - 1 (x is even). This is not an artificial limit, but an integral part of the construction of X[37]. The upshot is that any candidate j must be within one or two increments of the base value. A similar argument would hold for arrangements where x > y, except that the base value would be i. Examination of other combinations of x and y mod 8 show that solutions for k[9z] take the form 8^2(i^3 + k^3) + 8(3i^2s + 3k^2r) + (3is^2 + 3kr^2) + c = 8^2j^3 + 8(3)j^2r + 3jr^2 where 8i + s = x and 8k + r = y. For the particular parametric solution used for this experiment, c = 1 (1 mod 8) for s = 2, c = 27 (3 mod 8) for s = 6, c = 0 (0 mod 8) for s = 0, c = 8 (0 mod 8) for s = 2 leading to the congruence 8I + 3(is^2 + kr^2) + c == 8J + 3jr^2 0 + 3(is^2 + kr^2) + c == 0 + 3jr^2 r^2 == 1 mod 8 for the odd r 0 + 3(is^2 + k) + c == 0 + 3j the product is^2 == 0,4 mod 8 for the even s. Tabulating all the possibilities | s |s^2 |i mod 8 |c mod 8 |mod 8 0 == 3j - 3k | 0 == 3(3)(j - k) | 0 == (j - k) | 0 == (j - k) | 0,4 | 0 | all | 0 1 == 3j - 3k | 1(3) == 3(3)(j - k) | 3 == (j - k) | 3 == (j - k) | 0,4 | 0 | all | 1 (NA) 3 == 3j - 3k | 3(3) == 3(3)(j - k) | 9 == (j - k) | 1 == (j - k) | 0,4 | 0 | all | 3 (NA) 4 == 3j - 3k | 4(3) == 3(3)(j - k) | 12 == (j - k) | 4 == (j - k) | 2,6 | 4 | 1.3.5.7 | 0 (NA) 5 == 3j - 3k | 5(3) == 3(3)(j - k) | 15 == (j - k) | 7 == (j - k) | 2 | 4 | 1.3.5.7 | 1 7 == 3j - 3k | 7(3) == 3(3)(j - k) | 21 == (j - k) | 5 == (j - k) | 6 | 4 | 1.3.5.7 | 3 The entries marked "NA" represent combinations of s and c mod 8 that don't occur. The valid combinations (based on the value of c produced by particular combinations of s and r) are s^2 == 0, c == 0; s^2 == 4, c == 1; s^2 == 4, c == 3 In this particular example, s^2 == 2^2 == 4, 4i = 4(3) = 12 == 4, c = 1 == 1, so the candidate j would seem to equal 5 + 7 = 12, placing it out of range. In fact, the only possibility within range seems to make j = k, implying that z^3 = y^3 = x^3 + y^3, an impossibility. If this technique proves viable it can be extended to other odd powers. For the case of n^5 n^2(n + 3n + ... + (2n - 1)n) = n^2n^3 = n^5 = n^3 + 3n^3 + ... + (2n - 1)n^3 X[57], Z[57] would be X[57](x,y) = - 16x^3 - 7y^3 + 8x^4 + 14xy^3 + u[57] Z[57](z) = - 49z^3 + 14z^4 + v[57] with the linear equivalents X[57](x[1],x[2],x[3],x[4]) = - 16x[1] - 7x[2] + 8x[3] + 14 x[4] + u[57] Z[57](z[1],z[2]) = - 49z[1] + 14z[2] + v[57] The parametric solutions would be identical and the input values would have the same properties mod 8 as their respective partners in the third power problem, except that fourth and higher even powers of even integers are always 0 mod 8. Fourth and higher even powers of odd integers are still 1 mod 8. In terms of the possibilities mod 8 for k[9z] that were just tabulated, jr^2 would become jr^2m, m depending on the power n of the Fermat equation being considered. The coefficient on jr^2m would be the next-to-last coefficient in the binomial expansion for the proposed z^n. A work-up of Pascal's Triangle or the Binomial Formula shows that this coefficient always equals the power the binomial is being raised to. The coefficient for odd powers would therefore be 1, 3, 5, 7 mod 8 and the congruences would have 5(5)(j - k) == (j - k) or 7(7) (j - k) == (j - k). Reflecting on the nature of c, one might come to the conclusion that it must equal s^n/8. In order to be as concise as possible, it wasn't mentioned that the particular parametric solution used required a division by 8 in order to compute t[6] for the correct value of u[37]. Working this out further, this value for c is essential in order to make the parts of the working equation balance: the "missing" remainder 8 in the worked-out example is "displaced" into k[9z] and restored when 8k[9] + r[8] is evaluated. Testing with 34^5 + 43^5 34^5 + 43^5 = (8(5 679 424) + 32) + (8(18 376 025) + 243) = 8(24 055 449) + 275 = 192 443 867 The parametric solution once again yields the correct answer when r[8] = 275, but gives 8(24 055 453) + 243 = 192 443 624 + 243 = 192 443 867 = 34^5 + 43^5 for r[8] = 243, a difference of 4 = 2^5/8 = 32/8, as expected. Any proposed k[9] would now have to agree with 8(24 055 453) + 243 = (8j + r)^5 = 8(8^4j^5 + 8^3(5)j^4r + 8^2(10)j^3r^2 + 8(10)j^4r^3 + 5jr^4) + 3^5 = 8(8^4j^5 + 8^3(5)j^4r + 8^2(10)j^3r^2 + 8(10)j^4r^3 + 5jr^4) + 243 24 055 453 = 8^4j^5 + 8^3(5)j^4r + 8^2(10)j^3r^2 + 8(10)j^2r^3 + 5jr^4 = 4096j^5 + 2560j^4(3) + 640j^3(3^2) + 80j^2(3^3) + 5j3^4 = 4096j^5 + 7680j^4 + 5760j^3 + 2160j^2 + 405j j | sum | sum | mod 8 5 | 18376025 | 1 6 | 43128126 | 6 7 | 89365507 | 3 8 | 168765608 | 0 9 | 296630829 | 5 10 | 492380050 | 2 11 | 780040151 | 7 12 | 1188737532 | 4 8I + 5(is^4 + kr^4) + c == 8J + 5jr^4 0 + 5(is^4 + kr^4) + c == 0 + 5jr^4 r^4 == 1 mod 8 for the odd r, the product is^4 always == 0 For n = 5 the values of c: 0, (s = 0); 32/8 = 4 == 4, (s = 2); 1024/8 = 128 == 0, (s = 4); 7776/8 = 972 == 4, (s = 6) 0 + 5(0 + k) + 0 == 0 + 5j 0 + 5(0 + k) + 4 == 0 + 5j Tabulating | s | s^4 | i | c | mod 8 | mod 8 | mod 8 0 == 5j - 5k | 0 == 5(5)(j - k) | 0 == (j - k) | 0 == (j - k) | 0,4 | 0 | all | 0 4 == 5j - 5k | 4(5) == 5(5)(j - k) |20 == (j - k) | 4 == (j - k) | 2,6 | 0 | all | 4 For n = 7 the values of c: 0, (s = 0); 128/8 = 16 == 0, (s = 2); 16384/8 = 2048 == 0, (s = 4); 279936/8 = 34992 == 0, (s = 6). 0 + 7(0 + k) + 0 == 0 + 7j 0 == 7(j - k) 0 == 7(7)(j - k) 0 == (j - k) For n > 5, the only possibility seems to be that no candidates are feasible.