How to find first common point?

Here is something I really have tried to figure out,
but I am stumped by this problem.


Here it is (everything in integers):

1) There is two "sets"; A and B. Both are increasing in
the same way (we know the increasing).

2) B starts at a place on A (which we also know).

3) Sooner or later, they will share a common point (here called F).
Of course, there is infinite solutions, but it is all about
to find the FIRST one.

Let's look at a scenario with some examples.

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Scenario: The points in the two "sets" increases such, that

Point= X^2, so

If X= 0, 1, 2, 3, 4,... then

Points= 0, 1, 4, 9, 16,...

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

Example 1: B starts at A= 7

....01.4.....9......16
A: II--I--*-I------I
B:..........II--I----I
.............01.4....9

Their common point (F) is reached at A= 16

- - - - - - - -

Example 2: B starts at A= 11

....01..4...9.......16.......25..........36
A: II--I----I-*----I--------I----------I
B:...............II--I----I------I--------I
..................01.4....9......16.......25

Their common point (F) is reached at A= 36

- - - - - - - -

Example 3: B starts at A= 13

....01..4...9.......16.......25..........36............49
A: II--I----I---*--I--------I----------I------------I
B:..................II--I----I------I--------I----------I
.....................01.4....9......16........25.........36

Their common point (F) is reached at A= 49

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The formula for the scenario is: Start + K^2 = L^2

Here is a table showing the FIRST solution for every
"starting point" value. It goes like this:


Starting point...........Common point...........Start+K^2=L^2
-----------------.........--------------------.........------- --------------
B starts at A=3:.........B= 1, A= 4 -->..........3 + 1 = 2

B starts at A=5:.........B= 4, A= 9 -->..........5 + 2 = 3

B starts at A=7:.........B= 9, A= 16 -->.......7 + 3 = 4..........(Example 1)
B starts at A=8:.........B= 1, A= 9 -->.... ....8 + 1 = 3
B starts at A=9:.........B=16, A= 25 --> .......9 + 4 = 5

B starts at A=11:.......B=25, A= 36 -->.......11 + 5 = 6..........(Example 2)
B starts at A=12:.......B= 4, A= 16 -->........12 + 2 = 4
B starts at A=13:.......B=36, A= 49 -->.......13 + 6 = 7..........(Example 3)

B starts at A=15:.......B= 1, A= 16 -->.......15 + 1 = 4
B starts at A=16:.......B= 9, A= 25 --> .......16 + 3 = 5
B starts at A=17:.......B=64, A= 81 -->.......17 + 8 = 9

B starts at A=19:.......B=81, A=100 -->.......19 + 9 = 10
B starts at A=20:.......B=16, A= 36 -->........20 + 4 = 6
B starts at A=21:.......B= 4, A= 25 -->.........21 + 2 = 5

B starts at A=23:.......B121, A=144 -->.......23 + 11 = 12
B starts at A=24:.......B= 1, A= 25 -->.........24 + 1 = 5
B starts at A=25:.......B144, A=169 -->.......25 + 12 = 13

B starts at A=27:.......B= 9, A= 36 -->..........27 + 3 = 6
B starts at A=28:.......B=36, A= 64 -->.........28 + 6 = 8
B starts at A=29:.......B=196, A=225 -->.......29 + 14 = 15

B starts at A=31:.......B=225, A=256 -->.......31 + 15 = 16
B starts at A=32:.......B= 4, A=36 -->..........32 + 2 = 6
B starts at A=33:.......B=16, A=49 -->.........33 + 4 = 7

B starts at A=35:.......B=289, A=324 -->.......35 + 17 = 18
B starts at A=36:.......B=64, A=100 -->.........36 + 8 = 10
B starts at A=37:.......B=324, A=361 -->.......37 + 18 = 19

B starts at A=39:.......B=25, A=64 -->..........39 + 5 = 8
B starts at A=40:.......B= 9, A=49 -->...........40 + 3 = 7
B starts at A=41:.......B=400, A=441 -->.......41 + 20 = 21

B starts at A=43:.......B=441, A=484 -->.......43 + 21 = 22
B starts at A=44:.......B=100, A=144 -->.......44 + 10 = 12
B starts at A=45:.......B= 4, A=49 -->...........45 + 2 = 7

B starts at A=47:.......B=529, A=576 -->.......47 + 23 = 24
B starts at A=48:.......B=16, A=64 -->..........48 + 4 = 8
B starts at A=49:.......B=576, A=625 -->.......49 + 24 = 25


As we see, the answers ("Common point") are NOT
linear, because they are based on the fact that I want
the first common point...
It would be interesting to have a formula for solving this!


Thanks in advance!