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Nested radical redux (Posted on 2022-06-04)

Find all positive integers (x,y) that satisfy y=√(x+x√x)

Find all positive integers (x,y) that satisfy y=√(x+x√(x+x√x))

No Solution Yet Submitted by Jer

Rating: 5.0000 (2 votes)

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re: many more for part B

| Comment 6 of 9 |

(In reply to Parts A and B, and some OEIS by Larry)

See prior comment.

So with help from oeis A086340 Numbers n such that 1 + (n-1)*n*(n+1) is a square, I found that the sequence does not end at all after just the first 4 entries, but there is a large jump to the next one.

n values: [1,3,5,56,999800,2121328,2985696,3762725, ...]

{Addendum: this is in error, the entries greater than 56 are due to rounding error; see later comments}

So applying my function g(n), I got several more (x,y) pairs for part B:

(64, 40),

(576, 264),

(9828225, 1313565),

(999200239966002399920001, 999300174981000855552),

(20250292353302183164517889, 13903603856968828059648),

(79466181560370340717492225, 45989591503905574879232),

(200451779345854638707789376, 103337672651396498849792),

(365018169129726016397249289, 174592334534771629621248),

(711220293558142474393050625, 312970657702614147268608),

(1162260880699008904055833600, 480995117581527626547200),

(1206028458198029683169676801, 496807177219653293309952),

(1341739784760926249638016064, 545393252101684516093952),

(1935848725182298540541491201, 751644509983585937129472),

(2295223527835457320179456225, 872412468199841861730304),

(2360214910480210037698878729, 893989886913160908111872),

(2693874037647729796178679369, 1003644997266112860127232),

(3094972186005652053225611529, 1133247205086475700928512),

(3102403097744824140710150400, 1135627621551918886682624),

(3213530406643291208161380225, 1171142150416497815584768),

(3496065235561209579190461504, 1260759028413979370192896),

(4830226823492366448929652225, 1672906176736487211532288),

(4888864903168678604925795904, 1690662935809681432510464),

(5726513695771945383755602176, 1941574304569857585184768),

(6028890993162523175704957504, 2030989848253566956863488),

(6570517164602487212059453449, 2189775588153825230323712),

(7236713580848931221980185600, 2382861010336394189471744),

(7237278519447140400233102400, 2383023776643083724128256),

(8801212900443132984599116864, 2827969432249464696864768)

---------------

def g(n):

""" to generate x values for f """

return (n**2 - 1)**2

Edited on June 5, 2022, 6:26 am

Posted by Larry on 2022-06-04 14:12:03

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