Talk:Achilles number

Never consecutive?

Are there any consecutive Achilles numbers? --88.78.2.158 (talk) 16:29, 2 October 2009 (UTC)[reply]

Apparently so: see http://www.primepuzzles.net/problems/prob_053.htm which has several examples including 5425069447 and 5425069448 --Rumping (talk) 00:16, 3 March 2016 (UTC)[reply]

References

Are there any references to this concept in reliable sources? Sergeant Cribb (talk) 20:16, 31 May 2011 (UTC)[reply]

I would have thought OEIS and MathWorld were reliable sources --Rumping (talk) 00:16, 3 March 2016 (UTC)[reply]

Missing Achilles numbers?

27, 243, 343, 1331, 2187, 3375, 6859, 12167, 16807, 19683, 29791, 42875, 59319, 79507, 103823, 132651, 161051, 166375, 177147, ...

These are the numbers of the form (4k + 3)^{odd number}.

updated list of Achilles numbers up to 5000: 27, 72, 108, 200, 243, 288, 343, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1800, 1944, 2000, 2187, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3375, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000

smallest pair of consecutive Achilles numbers: 12167 = (4*5 + 3)³ and 12168 = 2³*3²*13² 2A00:6020:A123:8B00:74EC:9F2D:8C4D:2896 (talk) 20:41, 4 December 2023 (UTC)[reply]

(4k + 3)^{odd number} for an exponent above 1 is clearly a perfect power and therefore not an Achilles number. PrimeHunter (talk) 00:54, 5 December 2023 (UTC)[reply]