Tijdeman 1973 integers many small prime factors

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theorem_1: States that for positive integers 3 < a < b with r = omega(ab) and p = P(ab), b - a exceeds a/(log a)^C_1 with C_1 = c_1^(r^4) (log p)^(14r^2) for an effectively computable absolute c_1, and the corollary that the integers composed of primes at most p have gaps above n_i/(log n_i)^C(p).

theorem_2: States that for any set of r > 1 primes with largest element p there are infinitely many pairs a, b with 0 < b - a < (r log p)^r a/(log a)^(r-1), so the exponents in Theorem 1 and its corollary cannot be taken below r - 1 and pi(p) - 1.

theorem_3: States that for 0 < theta < 1 and positive integers a < b < a + a^(1-theta) with r = omega(ab) and p = P(ab), a < exp(((e^(4r^2) log p)/theta)^(7r^2)), which makes Erdős's threshold N_theta explicit.

theorem_4: States that positive integers a < b with r = omega(ab) and p = P(ab) satisfy a <= exp((10(b - a)p^r)^(10^5)), and the corollary that every a < b with b - a = k and P(ab) = p has log a at most (10ke^p)^(10^5).

theorem_5: States that positive integers a < b with the same greatest prime factor satisfy log(b - a) + P(a) >= 10^(-6) log log a and b - a >= 10^(-6) log log a.

theorem_6: States that if a < b <= a + a^(1-theta) with a = a_1 a_2, b = b_1 b_2, P(a_1 b_1) <= p and omega(a_1 b_1) <= r, then a_2 b_2 > a^eta - 1 for an effectively computable eta = eta(p, r, theta) > 0.

theorem_7: States that for each 0 < theta < 1 there is an infinite sequence of primes such that the integers composed of them satisfy n_(i+1) - n_i > n_i^(1-theta), which answers Wintner's question, Erdős Problem 240, in the affirmative.

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Robert Tijdeman, On integers with many small prime factors. Compositio Mathematica 26 (1973), no. 3, 319-330. The Numdam scan read for this card prints on its cover page "© Foundation Compositio Mathematica, 1973, tous droits réservés." and "Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.", every other right reserved.

Tijdeman uses the Gelfond-Baker method to make effective and nearly optimal the gap estimates for the sequence n_1 < n_2 < ... of integers composed only of primes up to p. Theorem 1 (p. 320) and its corollary (p. 321) give an effectively computable C = C(p) with n_{i+1} - n_i > n_i / (log n_i)^C for n_i >= 3, while Theorem 2 (p. 321) constructs infinitely many pairs showing the exponent cannot be taken below pi(p) - 1, so the gap between Erdos's bound n_{i+1} - n_i > n_i^{1-theta} and the opposite result n_{i+1}/n_i -> 1 is almost completely filled. Theorems 3 and 4 (pp. 322-323) give explicit effective bounds for the threshold N_theta in Erdos's inequality n_{i+1} - n_i > n_i^{1-theta} and for the thresholds A_{kp} above which consecutive such integers differ by more than k (yielding in Theorem 5, p. 324, for instance, b - a >= 10^{-6} log log a whenever a < b share the same greatest prime factor), Theorem 6 (p. 325) extends the estimate, at Straus's suggestion, to integers merely having many small prime factors rather than being composed entirely of them, and Theorem 7 (p. 326) answers a question of Wintner, Problem 240, in the affirmative by constructing, for each 0 < theta < 1, an infinite sequence of primes for which the integers built from them satisfy n_{i+1} - n_i > n_i^{1-theta}. Theorems 1, 3, 6 and 7 rest on lower bounds for linear forms in logarithms (Fel'dman's for Theorem 1, Baker's for the others, a then unpublished sharpening of Baker's for Theorems 6 and 7), Theorem 4 on Baker's bound for the integer solutions of y^2 = x^3 + k, Theorem 5 on the corollary of Theorem 4, and Theorem 2 on a pigeonhole argument.

Source: https://www.numdam.org/article/CM_1973__26_3_319_0.pdf.

Results

Page numbers are those of the journal print (pp. 319-330).

• Theorem 1 (p. 320) and its Corollary (p. 321): for positive integers 3 < a < b with r = omega(ab) and p = P(ab), b - a > a/(log a)^{C_1} with C_1 = c_1^{r^4}(log p)^{14r^2} and c_1 an effectively computable absolute constant; hence the integers n_i composed of primes at most p satisfy n_{i+1} - n_i > n_i/(log n_i)^C for n_i >= 3, with an effective C = C(p).
• Theorem 2 (p. 321): for a set of r > 1 primes with maximum p there are infinitely many pairs a, b with 0 < b - a < (r log p)^r a/(log a)^{r-1}; the proof takes a and b composed of those primes.
• Theorem 3 (p. 322): for 0 < theta < 1 and positive integers a < b < a + a^{1-theta} with r = omega(ab) and p = P(ab), a < exp{((e^{4r^2} log p)/theta)^{7r^2}}.
• Theorem 4 (p. 323) and its Corollary (p. 323): for positive integers a < b with r = omega(ab) and p = P(ab), a <= exp{(10(b-a)p^r)^{10^5}}; hence log A_{kp} <= (10 k e^p)^{10^5} for the least A_{kp} such that b - a = k and P(ab) = p force a <= A_{kp}.
• Theorem 5 (p. 324): for positive integers a < b with P(a) = P(b), log(b - a) + P(a) >= 10^{-6} log log a and b - a >= 10^{-6} log log a.
• Theorem 6 (p. 325): for 0 < theta < 1, a < b <= a + a^{1-theta}, a = a_1 a_2, b = b_1 b_2, P(a_1 b_1) <= p and omega(a_1 b_1) <= r, there is an effective eta = eta(p, r, theta) > 0 with a_2 b_2 > a^eta - 1.
• Theorem 7 (p. 326): for each 0 < theta < 1 there is an infinite sequence of primes p_1 < p_2 < ... whose integers n_1 < n_2 < ... satisfy n_{i+1} - n_i > n_i^{1-theta}.

Read status. Claims checked for the seven theorems and the two corollaries above, read clause by clause on the print; the proofs were read for their structure.

Bears on

• Problem 240: Theorem 7 gives, for each 0 < theta < 1, an infinite set of primes whose integers have gaps n_{i+1} - n_i > n_i^{1-theta}, which tend to infinity; this answers the problem's question in the affirmative. The paper presents it as solving Wintner's conjecture, cited from Erdos's 1965 survey (p. 326).
• Problem 1106: the paper does not mention partition numbers. Schinzel's proof that F(n) tends to infinity, printed by Erdos and Ivic (1990), cites it for the fact that two distinct integers composed of a fixed finite set of primes are not closer than A (log A)^{-C}, which Theorem 1 gives. The paper bears on neither a rate for F(n) nor the second question, whether F(n) > n for all large n.

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