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ramsey2879
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Let a triangular number T(n) = n*(n+1)/2 be factored into the product A*B with A less or equal to B.
Let gcd(x,y) be the greatest common divisor of x and y
For each of pair (A,B) define C,D,E,F as follows
C = (gcd(A,n+1))^2,
D = 2*(gcd(B,n))^2,
E = 2*(gcd(A,n))^2,
F = (gcd(B,n+1))^2.
As an example.
Let n = 36 then T(n) = 666 which can be factored into 6 distinct pairs A,B. The
six sets (A,B,C,D,E,F) are as follows
1. (1,666,1,648, 1369,2)
2. (2,333,1,162,1369,8)
3. (3,222,1,72,1369,18)
4. (6,111,1,18,1369,72)
5. (9,74,1,8,1369,162)
6. (18,37,1,2,1369,648)
My conjecture is that the products (A+Ct)*(B+Dt) and (A+Et)*(B+Ft) are each triangular numbers (= T(r) = r(r+1))/2) for all integer t. Also, the above example gave a interesting result in that when the 12 results for r (= the integer part of the square root of 2T(r)) are sorted by value the difference between adjacent r values when t = 1 was one of the even factors of 666. It would seem to me that the easiest way to prove this conjecture would be to derive and prove a formula for r. I will give the corresponding r values below for t = 1
1. 72,110
2. 54,184
3. 48,258
4. 42,480
5. 40,702
6. 38,1368
Also I found that r follows an arithmetic series as t varies, so it is important to find the pattern for the above values, since the difference between them and 36 is the difference value of the arithmetic series. Anyone see a pattern?
Let gcd(x,y) be the greatest common divisor of x and y
For each of pair (A,B) define C,D,E,F as follows
C = (gcd(A,n+1))^2,
D = 2*(gcd(B,n))^2,
E = 2*(gcd(A,n))^2,
F = (gcd(B,n+1))^2.
As an example.
Let n = 36 then T(n) = 666 which can be factored into 6 distinct pairs A,B. The
six sets (A,B,C,D,E,F) are as follows
1. (1,666,1,648, 1369,2)
2. (2,333,1,162,1369,8)
3. (3,222,1,72,1369,18)
4. (6,111,1,18,1369,72)
5. (9,74,1,8,1369,162)
6. (18,37,1,2,1369,648)
My conjecture is that the products (A+Ct)*(B+Dt) and (A+Et)*(B+Ft) are each triangular numbers (= T(r) = r(r+1))/2) for all integer t. Also, the above example gave a interesting result in that when the 12 results for r (= the integer part of the square root of 2T(r)) are sorted by value the difference between adjacent r values when t = 1 was one of the even factors of 666. It would seem to me that the easiest way to prove this conjecture would be to derive and prove a formula for r. I will give the corresponding r values below for t = 1
1. 72,110
2. 54,184
3. 48,258
4. 42,480
5. 40,702
6. 38,1368
Also I found that r follows an arithmetic series as t varies, so it is important to find the pattern for the above values, since the difference between them and 36 is the difference value of the arithmetic series. Anyone see a pattern?
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