Kulkarni 2005 class diophantine equations involving bernoulli polynomials

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theorem_1: States that for m >= n > deg(C) + 2 the equation a B_m(x) = b f_n(y) + C(y) has only finitely many rational solutions with bounded denominator, except in two explicit cases (m = n with m + 1 a perfect square, and m = 2n with (n + 1)/3 a perfect square), each with a uniquely determined C.

theorem_2: States that for m >= n > deg(C) + 2 the equation a f_m(x) = b B_n(y) + C(y) has only finitely many rational solutions with bounded denominator, except when m = n, m + 1 is a perfect square and b = a(sqrt(m+1))^m, where C is uniquely determined, given explicitly, and of degree m - 4.

theorem_c: Records the result the paper restates from Kulkarni and Sury (2003): if f_m(x) = g(y) has infinitely many rational solutions with bounded denominator, then g is f_m composed with a polynomial, or m is even and g factors through the product of (X - ((2i-1)/2)^2), or m = 4 and g has an explicit quadratic form.

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Manisha Kulkarni, B. Sury, A class of Diophantine equations involving Bernoulli polynomials. Indagationes Mathematicae (N.S.) 16 (1), 51-65 (2005). doi:10.1016/S0019-3577(05)80014-X.

Kulkarni and Sury study a B_m(x) = b f_n(y) + C(y) and a f_m(x) = b B_n(y) + C(y) for nonzero rationals a, b, a rational polynomial C, and m >= n > deg C + 2, where f_n(x) = x(x+1)...(x+n-1) and B_n is the nth Bernoulli polynomial. Theorem 1 (p. 52) shows the first equation has only finitely many rational solutions with bounded denominator except when m = n with m + 1 a perfect square and a = b(sqrt(m+1))^m, or m = 2n with (n+1)/3 a perfect square and a = b((n/2) sqrt((n+1)/3))^n; in each exceptional case a uniquely determined polynomial C gives infinitely many such solutions, with C identically zero when m = n = 3 and of degree n - 4 when n > 3. Theorem 2 (p. 52) is the analogous statement for the second equation, exceptional only when m = n with m + 1 a perfect square and b = a(sqrt(m+1))^m, with C then uniquely determined, given explicitly, and of degree m - 4. The remarks on p. 53 show the condition n > deg C + 2 is sharp. The method is the Bilu-Tichy criterion (Theorem A, p. 53) with the decomposition theorem for Bernoulli polynomials of Bilu, Brindza, Kirschenhofer, Pintér and Tichy (Theorem B, p. 54), worked out through explicit coefficient comparisons, one of them by a MAPLE computation (p. 57). For Theorem 2 the paper uses Theorem C (p. 61), which it restates from the authors' 2003 paper (Indag. Math. (N.S.) 14 (2003) 35-44) without proof: it lists the only cases in which f_m(x) = g(y) can have infinitely many rational solutions with bounded denominator. The paper proves nothing about problem 388 itself.

Source: https://doi.org/10.1016/S0019-3577(05)80014-X. No notice is printed on the publisher's PDF pages read; the article's registered DOI is 10.1016/S0019-3577(05)80014-X (10.1016/S0019-3577(05)80017-3 is not registered at Crossref), its publisher page could not be read on 2026-10-02 (the DOI resolved to a script-only redirect stub and ScienceDirect returned HTTP 403), its Crossref record lists only Elsevier's own text-and-data-mining and open-archive user licenses and no Creative Commons license, and ScienceDirect's site-wide footer (read 2026-10-02) reserves all rights, every other right reserved.

Bears on. #388: the problem's equation is f_{k_1}(m_1+1) = f_{k_2}(m_2+1) with k_1, k_2 > 3, and Theorem C, as the paper restates it on p. 61, says that for each fixed pair of lengths infinitely many solutions would put f_{k_2} in its case (1) or (2) (case (3) needs g of degree 2). It does not decide whether those cases occur, is not uniform in the lengths, and neither settles the finiteness question nor classifies the solutions. Theorems 1 and 2 concern Bernoulli polynomials and do not bear on the problem.

Results. Labels and pages are those of the print (pp. 51--65).

• Theorem 1 (p. 52): a B_m(x) = b f_n(y) + C(y), finitely many bounded-denominator rational solutions outside the two exceptional cases; the page also gives the remarks of p. 53, and its proof pointer the coefficient computation of p. 61.
• Theorem 2 (p. 52): a f_m(x) = b B_n(y) + C(y), finitely many outside m = n, m + 1 a perfect square, b = a(sqrt(m+1))^m.
• Theorem C (p. 61): restated from the authors' 2003 paper, not proved here; the necessary conditions for f_m(x) = g(y) to have infinitely many bounded-denominator rational solutions.

Theorems A and B (pp. 53--54) are cited background from Bilu and Tichy and from Bilu, Brindza, Kirschenhofer, Pintér and Tichy.

Read status. Claims checked for the three results above, read clause by clause on the print; the proofs were read for their structure only, and Theorem C was not checked against the 2003 paper.

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