Erdos 1985 my problems number theory i would

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conjecture_13: The conjecture, about 45 years old by the paper's account, that for an absolute constant c and every n the squared gaps between consecutive integers coprime to n sum to less than c n^2/phi(n), with its prime analogue (14).

question_p79_distinct_gaps: Asks for upper and lower estimates of h(x), the largest h such that for some n < x the h consecutive prime gaps from d_n on are all distinct, with the expectation h(x) > (log x)^alpha and the guess (12) that h(x)/log x -> 0.

question_p80_missing_gap: Defines r(x) as the smallest integer t for which d_n = t has no solution with n <= x, and records Erdős's expectation that r(x)/log x -> infinity, with his remark that even r(x) -> infinity cannot be attacked.

question_p80_totative_gaps: Asks to determine or estimate the smallest integer f(k) not of the form a_{i+1} - a_i, where the a_i are the integers in [1, n_k - 1] coprime to the product n_k of the first k primes.

theorem_p80_limit_points: Records the result, credited in the paper to Ricci and Erdős and proved by Brun's method, that the set of limit points of d_n/log n has positive measure, with the conjecture that d_n/log n is dense in (0, infinity).

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Paul Erdos, On some of my problems in number theory I would most like to see solved. Number Theory (Ootacamund, 1984), Lecture Notes in Mathematics 1122, Springer, 74-84 (1985). No notice is printed in the file (pp. 74--75 and 83--84 carry no copyright or license line); the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, read 2026-10-02, prints "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only."); the card records no DOI, so the publisher's chapter page was not consulted and no Crossref license is recorded; the term is unstated.

A short selection of the number theory problems Erdos most wanted solved, drawing on Guy's problem book and the Erdos-Graham monograph, and including consecutive prime differences, prime-counting inequalities and totatives of primorials. Among prime-gap questions he asks for estimates on h(x), the largest number of consecutive prime gaps d_n, ..., d_{n+h(x)-1} that are all distinct for some n < x, noting h(x) -> infinity follows from Brun's method and expecting h(x) > (log x)^a and possibly h(x)/log x -> 0. For #853 he defines r(x) as the smallest t for which d_n = t has no solution with n <= x, expects r(x)/log x -> infinity, and says even r(x) -> infinity cannot be attacked by available methods - the statement is made without the parity restriction the modern formulation needs. He also recalls his result with Ricci that the set of limit points of d_n/log n has positive measure while not a single finite limit point is known. For #854 he passes to the totatives 1 = a_1 < a_2 < ... < a_{phi(n_k)} = n_k - 1 of the primorial n_k, that is the integers below n_k with all prime factors exceeding p_k, and asks only to determine or estimate the smallest integer f(k) that is not of the form a_{i+1} - a_i, saying he has not done so; the entry's further abundance and maximum-gap questions are not in this paper. He ends that section with his roughly 45-year-old prize conjecture that sum (a_{i+1}-a_i)^2 over totatives of n is less than c n^2/phi(n), and the unreachable prime analog sum (p_{k+1}-p_k)^2 < c x log x; the next section recalls the Hensley-Richards theorem that pi(x+y) <= pi(x) + pi(y) is incompatible with the prime k-tuple conjecture.

Source: https://users.renyi.hu/~p_erdos/1985-17.pdf.

Read status. Claims checked: every statement on the result pages below was read clause by clause on the page images of the print (pp. 78--80). The paper proves nothing on these pages; the Ricci-Erdos theorem is recalled without proof or reference.

Results. question on h(x), with (12) (pp. 79--80); question on r(x) (p. 80); limit points of d_n/log n (p. 80, recalled); question on f(k) for the primorial (p. 80); Conjecture (13), with (14) (p. 80).

Bears on.

• #5: the recalled theorem of p. 80 gives a set of positive measure of limit points of d_n/log n without naming any one, so it decides no single value of C; the paper states that no finite limit point was known.
• #220: (13) on p. 80 is the problem's inequality as posed, recorded in the paper as open with a prize; the paper proves nothing on it.
• #852: the question of pp. 79--80 defines the problem's h(x); the expectation h(x) > (log x)^alpha and the guess (12) are the problem's two questions. The paper proves neither.
• #853: the problem's r(x), defined on p. 80 without the restriction to even t; the paper expects r(x)/log x -> infinity and calls even r(x) -> infinity beyond available methods. It proves neither.
• #854: the question of p. 80 is the problem's first part, without the restriction to even integers; the problem's comparison with the maximal gap is not in the paper. The paper records no answer.

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