{"schema":"vela.problem-packet.v0.1","problem":365,"statement":"Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations? In other words, must either $n$ or $n+1$ be a square?Is the number of such $n\\leq x$ bounded by $(\\log x)^{O(1)}$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A060355","name":"Numbers k such that k and k+1 are powerful numbers.","terms":"8,288,675,9800,12167,235224,332928,465124,1825200,11309768,384199200,592192224,4931691075,5425069447,13051463048,2213222","url":"https://oeis.org/A060355"},{"id":"A060859","name":"Powerful numbers of the form k^2 - 1.","terms":"8,288,675,9800,235224,332928,1825200,11309768,384199200,592192224,4931691075,13051463048,221322261600,443365544448,86536","url":"https://oeis.org/A060859"},{"id":"A175155","name":"Numbers m satisfying m^2 + 1 = x^2 * y^3 for positive integers x and y.","terms":"0,682,1268860318,1459639851109444,2360712083917682,86149711981264908618,4392100110703410665318,8171493471761113423918890","url":"https://oeis.org/A175155"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}