Cushing 2016 powerful numbers abc conjecture

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lemma_4_3: Cushing and Pascoe's Lemma 4.3: for a fixed k, n! + k is a powerful number for only finitely many n; the lemma's statement does not name the abc conjecture, but its proof assumes it, as Theorem 4.1 does.

theorem_4_1: Cushing and Pascoe's Theorem 4.1: for each k at least 0, assuming the abc conjecture, only finitely many powerful numbers lie within distance k of a factorial; the paper proves the cases n! and n! + k and leaves the case n! − k to the reader as Exercise 4.4.

theorem_5_2: Cushing and Pascoe's Theorem 5.2: assuming the abc conjecture, a coprime arithmetic progression contains only finitely many runs of three consecutive terms that are all powerful numbers.

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David Cushing, James Eldred Pascoe, Powerful numbers and the ABC-conjecture. arXiv:1611.01192 (2016). The copy read for this card is the arXiv preprint 1611.01192v1 (3 November 2016), whose title page prints the authors as David Cushing and J. E. Pascoe.

An expository note that states the abc conjecture twice, as Conjecture 1.1 (pp. 1--2, with $a,b,c$ pairwise coprime) and as Conjecture 3.1 (p. 4, with only $(a,b)=1$), each in the form that for every $\varepsilon>0$ only finitely many triples with $a+b=c$ have $\operatorname{rad}(abc)^{1+\varepsilon}<c$. It develops elementary properties of the radical (Lemmas 2.2, 2.3, 2.5, and Lemma 2.6, p. 3, which states $\operatorname{rad}(x)<x^{1/2}$ for powerful $x$, though its proof gives only $\operatorname{rad}(x)^2\le x$, with equality when $x$ is the square of a squarefree number, $x=1$ and $x=4$ among them), sketches Fermat's Last Theorem for large exponents under abc as a guided exercise (Example 3.2, pp. 4--5), and then applies the conjecture to powerful numbers. Theorem 4.1 (p. 5) states that for $k\ge0$, assuming abc, there are only finitely many powerful $x$ with $|x-n!|\le k$; the print leaves $n$ unquantified, and the introduction (p. 2) reads it as: for fixed $k$, $n!+k$ is powerful only finitely often, so powerful numbers lie near factorials only finitely often. The paper says it splits the proof into three lemmas; they are Lemma 4.2 ($n!$ itself is powerful only finitely often, unconditionally, from Bertrand's postulate), Lemma 4.3 ($n!+k$ is powerful only finitely often, by abc with $\varepsilon=\frac12$) and Exercise 4.4 ($n!-k$ is powerful only finitely often), which it leaves to the reader. Theorem 5.2 (p. 6) shows, assuming abc, that a coprime arithmetic progression with common difference $d$ (in the introduction, p. 2, $a_n=a+nd$ with $(a,d)=1$) contains only finitely many powerful triples $(a_k,a_{k+1},a_{k+2})$ (Definition 5.1, p. 6), generalizing the conditional result, which the introduction (p. 2) credits to its reference [3], that only finitely many triples of consecutive integers are all powerful; the introduction attributes that finiteness conjecture to Golomb and Erdős separately (p. 2), and Section 5 (p. 6) attributes to Erdős, Mollin and Walsh the conjecture that there are none. The method throughout is to bound the radical of a suitable abc triple and appeal to the conjecture. Section 6 (pp. 7--8) lists exercises, among them that $2^n+1$ is powerful only finitely often under abc (item 4, p. 7) and that $(n!)^r+k$ is powerful only finitely often (item 5, p. 8), neither proved; $2^n-1$ is not treated.

Source: https://arxiv.org/abs/1611.01192. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1611.01192), every other right reserved.

Read status: claims checked for the statements of Conjectures 1.1 and 3.1, Lemma 2.6, Theorem 4.1, Lemmas 4.2 and 4.3, Exercise 4.4, Definition 5.1, Theorem 5.2 and the Section 6 exercises, read clause by clause on the page images; the proofs were read for structure only, and nothing here is independently reviewed.

Bears on. #936: Theorem 4.1 with $k=1$ gives, assuming abc, that $n!+1$ and $n!-1$ are powerful for only finitely many $n$; the $n!+1$ case is proved as Lemma 4.3, and the $n!-1$ case rests on Exercise 4.4, left to the reader. That $2^n+1$ is powerful only finitely often is posed as an exercise and not proved, and $2^n-1$ is not treated. #364: Theorem 5.2 for the progression of positive integers gives, assuming abc, only finitely many triples of consecutive powerful integers, a finiteness statement that does not answer whether any such triple exists. #398: since squares are powerful, Lemma 4.3 with $k=1$ gives, assuming abc, only finitely many solutions of $n!+1=x^2$, which the introduction (p. 2) recalls as Overholt's earlier result under abc; it does not answer whether the known solutions are the only ones.

Results.

• Theorem 4.1 (p. 5): for $k\ge0$, assuming abc, finitely many powerful $x$ with $|x-n!|\le k$; the case $n!-k$ rests on Exercise 4.4, left to the reader. Lemma 4.2 (p. 5) and Lemma 2.6 (p. 3) are recorded on that page.
• Lemma 4.3 (p. 5): $n!+k$ is powerful only finitely often, proved from abc.
• Theorem 5.2 (p. 6): assuming abc, a coprime arithmetic progression contains only finitely many powerful triples.

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