3.2 Intersection of Two Integer Circles with different radii

Here we denote the intersection of two integer circles by

𝒮⁢(ld⁡(c1,c2);r1,r2),\mathcal{S}(\operatorname{ld}(c_{1},c_{2});r_{1},r_{2}),

where r1r_{1} and r2r_{2} are the radii of the integer circles and ld⁡(c1,c2)\operatorname{ld}(c_{1},c_{2}) is the integer distance between their centers. For computations of densities, we may assume the centres are fixed at particular co-ordinates.

Theorem 3.2.1.

Consider the intersection 𝒮⁢(a;b,c)\mathcal{S}(a;b,c). Then this intersection is empty precisely when

gcd⁡(a,b)≠gcd⁡(b,c).\gcd(a,b)\neq\gcd(b,c).
Proof.

Suppose as a contradiction there is a point (x,y)(x,y) in this intersection.

Then gcd⁡(x,y)=b\gcd(x,y)=b and gcd⁡(x−a,y)=c\gcd(x-a,y)=c. From the first equation, we there exists integers x^,y^\hat{x},\hat{y} such that x=x^⁢bx=\hat{x}b and y=y^⁢by=\hat{y}b.Thus we rewrite the second statement

gcd⁡(x^⁢b−a,y^⁢b)=c.\gcd(\hat{x}b-a,\hat{y}b)=c.

Let AA be the greatest common divisor of aa and bb. Then b=b^⁢Ab=\hat{b}A and a=a^⁢Aa=\hat{a}A. So

gcd⁡(A⁢x^⁢b^−A⁢a^,A⁢y^⁢b^)=c.\gcd(A\hat{x}\hat{b}-A\hat{a},A\hat{y}\hat{b})=c.

Thus

c=A⁢gcd⁡(x^⁢b^−a^,y^⁢b^).c=A\gcd(\hat{x}\hat{b}-\hat{a},\hat{y}\hat{b}).

Thus

gcd⁡(b,c)=A⁢gcd⁡(b^,gcd⁡(x^⁢b^−a^,y^⁢b^)).\gcd(b,c)=A\gcd(\hat{b},\gcd(\hat{x}\hat{b}-\hat{a},\hat{y}\hat{b})).

Since gcd⁡(a^,b^)=1\gcd(\hat{a},\hat{b})=1, the previous equation implies that

gcd⁡(b,c)=A=gcd⁡(a,b).\gcd(b,c)=A=\gcd(a,b).

Thus we have found a contradiction. ∎

Let us conclude with the following conjecture:

Conjecture 1.

We have the following ratio:

#ℐ⁢𝒮⁢(a;1,1)=1a2⁢#ℐ⁢𝒮⁢(1;a,1).\#_{\mathcal{I}}\mathcal{S}(a;1,1)=\frac{1}{a^{2}}\#_{\mathcal{I}}\mathcal{S}(% 1;a,1).

This conjecture is supported by the following numerical experiments:

Example 3.2.2.

Let us consider #ℐ⁢𝒮⁢(a;1,1)\#_{\mathcal{I}}\mathcal{S}(a;1,1) and #ℐ⁢𝒮⁢(1;a,1)\#_{\mathcal{I}}\mathcal{S}(1;a,1). Using a computer algorithm in Appendix F, we calculated the ratio between our approximations. The results are recorded in the following table. Here:

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    ​

    aa is the different values of aa used for our computation;

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    ​

    Ratio is the ratio #ℐ⁢𝒮⁢(1;a,1)#ℐ⁢𝒮⁢(a;1,1)\frac{\#_{\mathcal{I}}\mathcal{S}(1;a,1)}{\#_{\mathcal{I}}\mathcal{S}(a;1,1)};

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    ​

    Continued Fraction is the continued fraction form of the ratio;

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    ​

    Estimate is formed by removing a large term from the continued fraction, as this is likely an error in the approximation.

a Ratio Continued Fraction Estimate
3 9.025828588 9; 38, 1, 2, 1, 1, 7, 1, 1, 2, 1, 1, 1, 1, 1, 4, 6, 15, 1 [9] = 9 = 32
4 15.98852653 15; 1, 86, 6, 2, 1, 8, 2, 2, 1, 2, 31, 2, 1, 3 [15;1] = 16 = 42
5 25.0134942 25; 74, 9, 2, 3, 1, 3, 1, 6, 2, 3, 3 [25] = 25 = 52
6 35.97176004 35; 1, 34, 2, 2, 3, 3, 2, 1, 6, 4, 2, 1, 2, 7, [35;1] = 36 = 62
7 49.05757858 49; 17, 2, 1, 2, 1, 1, 2, 1, 2, 13, 2, 23, 1, 1 [49]=49 = 72
8 63.98140626 63; 1, 52, 1, 3, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 4, 1, 2, 2 [63;1]=64=82
9 81.37311128 81; 2, 1, 2, 7, 1, 8, 1, 3, 1, 1, 1, 3, 1, 9, 1, 4, 1, 1, 2 No large removable terms
10 99.35671451 …
11 120.4585646
12 145.0706422
13 169.8734475
14 196.6579863
15 223.9634909
16 254.1930446
17 289.192548
18 320.5415581
19 361.1491913
20 399.4738194
21 439.5027705
22 489.8939052
23 535.8277433
24 565.8133947
25 619.3000919
26 685.2742616
27 714.6634383
28 775.7552941
29 856.3562914
30 915.1652139

The table above shows a that for small values of aa, the ratio is very close to a2a^{2}, but this form is not supported for values of aa greater than or equal to 9. To see if this is due to error our my approximations or a change in formula from this point, we reran the algortihm for a∈[9,30]∩ℤa\in[9,30]\cap\mathbb{Z} with a later term in the filtration ℐ\mathcal{I}.

r Ratio Continued Fraction
9 80.86013465 80; 1, 6, 6, 1, 2, 9, 27, 1, 1, 5, 3, 1, 11
10 99.94571594 99; 1, 17, 2, 2, 1, 2, 4, 1, 1, 3, 3, 1, 1, 49, 11
11 120.9047662 120; 1, 9, 1, 1, 527, 2, 2, 10, 1, 7
12 143.8853871 143; 1, 7, 1, 2, 1, 1, 1, 3, 28, 1, 138, 1, 6
13 168.5743996 168; 1, 1, 2, 1, 6, 6, 2, 5, 1, 15, 39
14 195.9302836 …
15 225.4916487
16 255.5628954
17 289.1446574
18 324.4596464
19 361.4093346
20 399.1691446
21 441.2575198
22 484.475851
23 529.7113125
24 577.1792046
25 625.7090311
26 676.2501771
27 729.7783561
28 783.6976088
29 844.530839
30 898.8380365

The larger sample size with large values of aa does not come as close to the predicted value as the smaller values of aa did, but the values do get closer. This provides a good enough basis for us to guess that Conjecture 1 is correct as a starting point for further study.