a = √(√(n) + √((k*n)+1))

 
 
Can you find integer solutions of

KNUTH 1

 
 
This can be viewed as a Pell Equation

a^2 \; = \; \sqrt {n} \; + \; \sqrt {(k \, n) + 1}

in integers, so we know that there exist integers   x   and   y   such that:

n \; = \; y^2
k \, n \; + \; 1 \; = \; x^2
x \; + \; y \; = \; a^2

k \, y^2 \; + \; 1 \; = \; x^2

 
 
 
 

 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , . Bookmark the permalink.

2 Responses to a = √(√(n) + √((k*n)+1))

  1. paul says:

    There are an infinite number of solutions to that. There are infinite sets when n=1 and n=4 and n=9, it seems to miss 16, back at 25 and 36. There is an interesting relationship at n=36, the value of “a” is defined by the pairs of 18t+-5, which goes {{5}, {13, 23}, {31,41},{49,59},{67,77},{85,95},…}
    Here are some values at n=36, format is {n, k, a}

    {36,10,5}
    {36,738,13}
    {36,7598,23}
    {36,25334,31}
    {36,77934,41}
    {36,159334,49}
    {36,335434,59}
    {36,558258,67}
    {36,974498,77}
    {36,1447610,85}

    {36,288372401010,1795}..t=100
    {36,294852439010,1805}..t=100

    The last two were found by solving back for k

    Paul.

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