Every sufficiently large even number is the sum of two primes

The binary Goldbach conjecture asserts that every even integer greater than $4$ is the sum of two primes. In this paper, we prove that there exists an integer $K_\alpha$ such that every even integer $x > p_k^2$ can be expressed as the sum of two primes, where $p_k$ is the $k$th prime number and $k > K_\alpha$. To prove this statement, we begin by introducing a type of double sieve of Eratosthenes as follows. Given a positive even integer $x > 4$, we sift from $[1, x]$ all those elements that are congruents to $0$ modulo $p$ or congruents to $x$ modulo $p$, where $p$ is a prime less than $\sqrt{x}$. Therefore, any integer in the interval $[\sqrt{x}, x]$ that remains unsifted is a prime $q$ for which either $x-q = 1$ or $x-q$ is also a prime. Then, we introduce a new way of formulating a sieve, which we call the sequence of $k$-tuples of remainders. By means of this tool, we prove that there exists an integer $K_\alpha > 5$ such that $p_k / 2$ is a lower bound for the sifting function of this sieve, for every even number $x$ that satisfies $p_k^2 < x < p_{k+1}^2$, where $k > K_\alpha$, which implies that $x > p_k^2 \; (k > K_\alpha)$ can be expressed as the sum of two primes.Comment: 32 pages. The manuscript was edited for proper English language by one editor at American Journal Experts (Certificate Verification Key: C0C3-5251-4504-E14D-BE84). However, afterwards some changes have been made in sections 1, 6, 7 and

Efficient long division via Montgomery multiply

We present a novel right-to-left long division algorithm based on the Montgomery modular multiply, consisting of separate highly efficient loops with simply carry structure for computing first the remainder (x mod q) and then the quotient floor(x/q). These loops are ideally suited for the case where x occupies many more machine words than the divide modulus q, and are strictly linear time in the "bitsize ratio" lg(x)/lg(q). For the paradigmatic performance test of multiword dividend and single 64-bit-word divisor, exploitation of the inherent data-parallelism of the algorithm effectively mitigates the long latency of hardware integer MUL operations, as a result of which we are able to achieve respective costs for remainder-only and full-DIV (remainder and quotient) of 6 and 12.5 cycles per dividend word on the Intel Core 2 implementation of the x86_64 architecture, in single-threaded execution mode. We further describe a simple "bit-doubling modular inversion" scheme, which allows the entire iterative computation of the mod-inverse required by the Montgomery multiply at arbitrarily large precision to be performed with cost less than that of a single Newtonian iteration performed at the full precision of the final result. We also show how the Montgomery-multiply-based powering can be efficiently used in Mersenne and Fermat-number trial factorization via direct computation of a modular inverse power of 2, without any need for explicit radix-mod scalings.Comment: 23 pages; 8 tables v2: Tweak formatting, pagecount -= 2. v3: Fix incorrect powers of R in formulae [7] and [11] v4: Add Eldridge & Walter ref. v5: Clarify relation between Algos A/A',D and Hensel-div; clarify true-quotient mechanics; Add Haswell timings, refs to Agner Fog timings pdf and GMP asm-timings ref-page. v6: Remove stray +bw in MULL line of Algo D listing; add note re byte-LUT for qinv_