Determine all pairs **(n, k)** of integers such that **0 < k < n** and

__Solution__ :

Here any integer value of will yield nonnegative integer values of and ;

however, the condition requires that

and

are each at least 2.

Hence the values are excluded,

while every and

will yield permissible values for

which satisfy the equation.

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There are two sets of recurrences for {n, k} both are defined with the same recurrence numbers but on different series, the recurrences are {3, -3, 1} and the generating numbers are:-

{7, 14, 23, 34…} and {2, 5, 9, 14..}. They are set up as follows

{7, 2}, {7, 5}

{14, 5},{14,9}

{23, 9}, {23,14}

{34, 14},{34,20}

…

…

The extended series are

n = {7,14,23,34,47,62,79,98,119,142,167,194,223,254,287,322,359,398,439,482}

k = {2,5,9,14,20,27,35,44,54,65,77,90,104,119,135,152,170,189,209,230}

The first few terms are

7 + 35 = 42

35 + 7 = 42

1001 + 3003 = 4004

3003 + 1001 = 4004

490314 + 1144066 = 1634380

1144066 + 490314 = 1634380

927983760 + 1855967520 = 2783951280

1855967520 + 927983760 = 2783951280

6973199770790 + 12551759587422 = 19524959358212

12551759587422 + 6973199770790 = 19524959358212

There are an infinite number.

Paul.

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