Gyory 1986 prime factors sums integers i

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corollary_1: Győry, Stewart and Tijdeman's corollary that for finite sets A and B of positive integers with |A| >= |B| >= 2 and k = |A|, some a in A and b in B have P(a + b) > C_7 log k log log k, with C_7 effectively computable.

theorem_1: Győry, Stewart and Tijdeman's theorem that for finite sets A and B of positive integers with |A| >= |B| >= 2, the product of all sums a + b has more than C_4 log |A| distinct prime factors, with C_4 an effectively computable positive constant.

theorem_2: Győry, Stewart and Tijdeman's theorem that for positive integers a_1 < ... < a_k and b with gcd(a_1, ..., a_k, b) = 1, the greatest prime factor of a_1 ... a_k (a_1 + b) ... (a_k + b) exceeds min((1 - eps) k log k, C_8 log log(a_k + b)) for k > k_0(eps), and that P(a_1 a_2 (a_1 + b)(a_2 + b)) tends to infinity with a_2 + b.

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Kálmán Győry, Cameron L. Stewart, Robert Tijdeman, On prime factors of sums of integers I. Compositio Mathematica 59 (1986), 81-88. The file prints "© Foundation Compositio Mathematica, 1986, tous droits réservés.", every other right reserved.

Theorem 1 shows that for finite sets A and B of positive integers with |A| >= |B| >= 2 and k = |A|, the number of distinct prime factors of the product of all sums a + b exceeds C_4 log k, with C_4 an effectively computable positive constant; this proves a conjecture of Erdős and Turán (stated for two sets of the same size k at least f(w)) with f(w) = e^{C_3 w}, and needs only that the second set have at least two elements. The result covers the 1934 Erdős-Turán inequality for a single set as a special case. Corollary 1 combines Theorem 1 with the prime number theorem to give a in A and b in B whose greatest prime factor P(a + b) exceeds C_7 log k log log k, extending to two sets the consequence the Erdős-Turán bound gives for one set. The proof of Theorem 1 uses Evertse's bound (Lemma 4, p. 84) of at most 3*7^{2w+3} coprime solutions of lambda x + mu y = nu z with x, y, z composed of w given primes, a result proved by a modification of the Thue-Siegel hypergeometric method; the original Erdős-Turán proof was elementary, and the elementary argument of Stewart and Tijdeman in part II of the series (cited as to appear) gives only C_5 (log l)/log log l with l = |B|. Theorem 2 improves (3) of Corollary 1 when some sums a + b are sufficiently large and all the sums have greatest common divisor one: for eps > 0 and k >= 2, if a_1 < ... < a_k and b are positive integers with gcd(a_1, ..., a_k, b) = 1, then P(a_1 ... a_k (a_1 + b) ... (a_k + b)) exceeds min((1 - eps) k log k, C_8 log log(a_k + b)) for k > k_0(eps), and P(a_1 a_2 (a_1 + b)(a_2 + b)) tends to infinity with a_2 + b over a_1 < a_2 with gcd(a_1, a_2, b) = 1. For problem 126, Theorem 1 with B = A gives more than C_4 log k distinct prime factors for the product of a + b over all pairs a, b in A, which also contains the terms 2a, and the paper notes that it covers the Erdős-Turán bound (1) for that product; the problem's product runs over a != b, and the paper proves no bound of larger order than log k.

Source: http://www.numdam.org/item/CM_1986__59_1_81_0/.

Bears on. #126: Theorem 1 (p. 81) with $B=A$ gives $\omega\bigl(\prod_{a,b\in A}(a+b)\bigr)>C_4\log k$, a product that also contains the terms $2a$, while the problem counts the prime factors of the product over $a\ne b$ and asks for growth faster than $\log n$; the paper proves no bound of larger order than $\log k$ and does not address the problem's question.

Results.

• Theorem 1 (p. 81): for finite sets $A$, $B$ of positive integers with $\lvert A\rvert\ge\lvert B\rvert\ge2$ and $k=\lvert A\rvert$, $\omega\bigl(\prod_{a\in A,b\in B}(a+b)\bigr)>C_4\log k$.
• Corollary 1 (p. 82): under the same hypotheses some $a\in A$, $b\in B$ have $P(a+b)>C_7\log k\log\log k$.
• Theorem 2 (pp. 82--83): for $\epsilon>0$, $k\ge2$ and positive integers $a_1<\cdots<a_k$ and $b$ with $\gcd(a_1,\ldots,a_k,b)=1$, $P\bigl(a_1\cdots a_k(a_1+b)\cdots(a_k+b)\bigr)>\min\bigl((1-\epsilon)k\log k,C_8\log\log(a_k+b)\bigr)$ for $k>k_0(\epsilon)$, and $P\bigl(a_1a_2(a_1+b)(a_2+b)\bigr)\to\infty$ as $a_2+b\to\infty$ with $a_1<a_2$ and $\gcd(a_1,a_2,b)=1$.

Read status: claims checked for Theorems 1 and 2, Corollary 1 and Lemma 4, read clause by clause on the page images of the print; the proofs of Theorem 1 and Theorem 2 followed, the latter in outline. Lemmas 1 to 4 are cited from other papers and were not checked. Nothing here is independently reviewed.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.