Applications of the abc Conjecture to Powerful Numbers

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P. A. CrowdMath, Applications of the abc conjecture to powerful numbers. arXiv:2005.07321 (2020). The copy read for this card is arXiv v1, submitted 15 May 2020 and the only version; its PDF is dated May 18, 2020.

Assuming the abc conjecture, the paper answers four questions of Cushing and Pascoe about powerful numbers (integers n with p^2 | n whenever p | n). Theorem 2.1 shows that x + y = z holds for only finitely many triples of 4-powerful numbers x, y, z with gcd(x, y) = 1, taking a = x, b = y, c = z and epsilon = 1/3 in the abc inequality. Theorem 2.2 shows that, for n >= 5, only finitely many powerful z arise as z = x^n + y^n with x and y coprime, using epsilon = 1/9 and the estimate (xy)^{10} < z^4; the cases n = 2 and n = 3 have infinitely many such powerful values (from Pythagorean triples and x^3 + y^3 = z^2) and n = 4 remains open, so Problem 3 of Cushing-Pascoe is only partially resolved. The paper further proves that for fixed coprime positive integers k, r only finitely many powerful numbers occur among the values k^n + r (Problem 4) and that (n!)^r + k is powerful for at most finitely many n (Problem 5), plus an additional CrowdMath forum problem (Theorem 2.6: the gaps between consecutive 3-powerful numbers tend to infinity), closing with a list of open problems. All results are conditional on abc, which is the relevant caveat for the powerful-number question of problem 936. For that problem, Theorem 2.3 with k = 2, r = 1 covers 2^n + 1 and Theorem 2.4 with r = k = 1 covers n! + 1; both theorems need the added constant positive, so the paper treats neither 2^n - 1 nor n! - 1.

Source: https://arxiv.org/abs/2005.07321. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2005.07321), every other right reserved.

Bears on. #936

Results to transcribe.

• Theorem 2.1: Assuming abc, x+y=z holds for only finitely many triples of 4-powerful numbers x, y, z with gcd(x,y)=1.
• Theorem 2.2: Assuming abc, for n ≥ 5 only finitely many powerful z arise as z = x^n + y^n with x, y coprime; n=2,3 admit infinitely many and n=4 is open.
• Theorem 2.3 (Problem 4): Assuming abc, if k and r are fixed coprime positive integers, only finitely many powerful numbers occur among the values k^n + r.
• Theorem 2.4 (Problem 5): Assuming abc, for each fixed choice of positive integers r and k, (n!)^r + k is powerful for at most finitely many n.

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