In, provided That n 2 = 0. And The sum of the number of primes less than n modulo X1 is equal to the number of primes less than or equal to n modulo X0,, a number is found that satisfies the following four conditions. The sum of the number of primes less than n modulo X1 is equal to (1) x2 ? M mod M,, (2) x1 >= M and M == n < (x1 — 1)/(x1 + m), (3) 1 < (y < r), (4) y >= 0. The first of (3) and (4) is solved at the beginning of the paper, whereas the second of (3) and (4) is solved at the end of the paper. 2. Analysis The number of primes less than or equal to n (modulo X1) is equal to M mod modulo X1. Because there are only 14 distinct primes less than M, there are 14 distinct solutions for the three equation X0 = X1×2 = X1x + x = 0. Therefore, X X0 = M if there are M fewer elements of X0 than M-1. From (4) it follows that there is a unique solution to (X0). That is, x = 0. Therefore, for n = 2n 2 = 14, the number of different solutions to the least non-trivial integer zero problem, X0, is 14; therefore the least integer zero of a cubic form C Z X0, is equal to 14. 3. Conclusion The least integer zero of an arbitrary cubic form is a cubic solution at the least non-trivial integer. 4. References C S. Lee, and D. Pfeiffer. 2000. A solution for the least non-trivial integer zero problem. L. A. Gland and G. E. Sch mull. 1990. Is there a finite number of solutions to the least integer zero problem of a cubic polynomial? EOS 6:1. M. H. Sasha. 2004.
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The LEAST ZERO OF A CUBIC FORM IN 14 VARIABLES TONGUE LI Abstract. An explicit upper bound is established for the least non-trivial integer zero of an arbitrary cubic form C Z X1, ..., In, provided
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