A New Method in the Problem of Three Cubes

In the current paper we are seeking P1(y),P2(y),P3(y) with the highest possible degree polynomials with integer coefficients, and Q(y) via the lowest possible degree polynomial, such that P1(y)^3+P2(y)^3+P3(y)^3=Q(y). Actually, the solution of this problem has close relation with the problem of the sum of three cubes a^3+b^3+c^3=d, since deg Q(y)=0 case coincides with above mentioned problem. It has been considered estimation of possibility of minimization of deg Q(y). As a conclusion, for specific values of d we survey a new algorithm for finding integer solutions of a^3+b^3+c^3=d.

1. Some facts on the history of the equation a 3 +b 3 +c 3 = d The question as to which integers are expressible as a sum of three integer cubes is over 160 years old. The first known reference to this problem was made by Fermat, who has offered to find the three nonzero integers, so that the sum of their nth powers is equal to zero. This framework of the problem makes the beginning of survey of equation a 3 + b 3 + c 3 = d for many mathematicians.
We present some significant result schedule. 1825 year -S. Ryley in [1] gave a parametrization of rational solutions for d ∈ Z: 1908 year -A.S. Werebrusov [2] found the following parametric family for d = 2: 1936 year -Later in [3] Mahler discovered a first parametric solution for d = 1: 1942 year -Mordell proved in [2] that for any other d a parametric solution with rational coefficients must have degree at least 5. 1954 year -Miller and Woollett [4]  1992 year -the first solution for d = 39 was found. Heath-Brown, Lioen, and te Riele [8] determined that 39 = 134476 3 + 117367 3 + (−159380) 3 with the rather deep algorithm of Heath-Brown [6]. This algorithm involved searching for solutions for a specific value of d using the class number of to eliminate values of a, b, c which would not yield a solution.
1994 year -Koyama [7] used modern computers to expand the search region to |a|, |b|, |c| 2 21 and successfully found first solutions for 16 integers between 100 and 1000 [9]. Also in 1994, Conn and Vaserstein [8] chose specific values of d to target, and then used relations implied by each chosen value to limit the number of triples (a, b, c) searched. So doing, they found first representations for 84 and 960. Their paper also lists a solution for each d < 100 for which a representation was known.
1995 year -Bremner [9] devised an algorithm which uses elliptic curve arguments to narrow the search space. He discovered a solution for 75 (and thus a solution for 600), leaving only five values less than 100 for which no solution was known. Lukes then extended this search method to also find the first representations for each of the values 110, 435, and 478 [10].
1997 year -Koyama, Tsuruoka, and Sekigawa [11] used a new algorithm to find first solutions for five more values between 100 and 1000 as well as independently finding the same solution for 75 that Bremner found. Also in the same paper, the authors discuss the complexity of the above algorithms.
1999 year -Bernstein [12] had implemented the method of Elkies [13]  It may be worth mentioning that the complete rational-solution of the equation a 3 + b 3 + c 3 = d 3 is known, and is given by where q, x, y are any rational numbers. So if we set q equal to the inverse of we have rational solutions of (2.1).
However, the problem of finding the integer-solutions is more difficult. If d is allowed to be any integer (not just 1) then Ramanujan gave the integer This occasionally gives a solution of equation ( There are known to be infinitely many algebraic solutions, for example However, it's not known whether every solution of the equation lies in some family of solutions with an algebraic parameterization.
Interestingly, note that if you replace 1 by 2, then again there's a parametric solution: but again this doesn't cover all known integer solutions. Note, that precisely one solution is known that is not given by (2.3) (see [16]): It's evidently not known up todays if there are any other algebraic solutions besides the one noted above.
In general it seems to be a difficult problem to characterize all the solutions of for some arbitrary integer d > 2. In particular, the question of whether all integer solutions are given by an algebraic identity seems both difficult and interesting.
Note that for d ≡ ±4(mod9) there are no solutions since, for any integer a, a 3 ≡ 0, 1, −1(mod9). It is a long standing problem as to whether every rational integer d = 4, 5( mod 9) can be written as a sum of three integral cubes.
According to the web page [12] of Daniel Bernstein, the first attacks by computer were carried out as early as 1955.  Consider now some specific: d = m 3 , d = m 12 and d = 2m 9 type values of d.
Multiply both sides of (2.3) by m 9 , and apply the change of variable t → t/m to obtain the more general solution 3 gives a primitive solution. For example, for l, k 1 the are primitive for GCD(3t, 2m) = 1, GCD(2t, 3m) = 1 and GCD(t, 6m) = 1 respectively.

New method and results
In this section a new method and results are surveyed. Here we consider more general framework of the problem of sums of three cubes. We are seeking P 1 (y), P 2 (y), P 3 (y) with the highest possible degree polynomials with integer coefficients and Q(y) with the lowest possible degree polynomial, so that Actually the solution of this problem has close relation with the above trivial problem, since the case of deg Q(y) = 0 coincides with our problem. Nevertheless the estimation of possibility of minimization of deg Q(y) itself is also an interesting problem.
RESULT 1. The first result of this paper is devoted to the case of degrees (8,8,6). We search the desired polynomials within the class of polynomials of the form and obtain the form Further considerations are devoted to the finding of cases interesting for us.
The result obtains the form A 12 729a 9 , which is a cube of an integer, so it is primitive (not interesting).
The result: Factor the last term, then the result obtains the form We do the substitution a = 3A 3 2b 2 . Then the result will get the form: (desirably less than 1000).