For what integers (x,y)
is 1/22
expressible as ( 1/x + 1/y ) ?
On 8/29/2022 8:53 PM, henh...@gmail.com wrote:
For what integers (x,y)
is 1/22 expressible as ( 1/x + 1/y ) ?
[spoiler space]
Both x and y must be non-zero. (If x is zero, then 1/x is infinitely
large, so 1/x + 1/y is also infinitely large. Similar for y.)
At least one of x, y must be positive. (If both are negative, then 1/x
and 1/y are also both negative, so 1/x + 1/y is also negative.)
Without loss of generality, assume that x > 0 and x >= y. (Each
resulting pair can be swapped.) Then consider:
1/22 = 1/x + 1/y
xy = 22y + 22x
(x - 22)y = 22x
y = 22x/(x - 22)
Now we can just look at x = 1, 2, 3, etc. and see which ones result in
y being an integer:
11, -22 -> 22/(22*11) - 11/(22*11)
18, -99 -> 22/(22*18) - 4/(22*18)
20, -220 -> 22/(22*20) - 2/(22*20)
21, -462 -> 22/(22*21) - 1/(22*21)
23, 506 -> 22/(22*23) + 1/(22*23)
24, 264 -> 22/(22*24) + 2/(22*24)
26, 143 -> 22/(22*26) + 4/(22*26)
33, 66 -> 22/(22*33) + 11/(22*33)
44, 44 -> 22/(22*44) + 22/(22*44)
note that 22 is a product of 2 primes
i was wondering if we replace the 22 with another product of 2 primes
we'd get the same pattern of 9 (=5+4) answer-pairs.
On 9/5/2022 5:59 PM, henh...@gmail.com wrote:
note that 22 is a product of 2 primes
i was wondering if we replace the 22 with another product of 2 primesI think we would.
we'd get the same pattern of 9 (=5+4) answer-pairs.
Let z = x - 22, then
y = 22(z + 22)/z
= 22z/z + (22^2)/z
= 22 + (22^2)/z
So the potential solutions are those where z is a factor of
22^2 = 2^2 * 11^2
or else the additive inverse of such a factor.
Similarly, if we replace 22 with the product of any two distinct
primes p and q, then x - pq must be of the form
(-1)^a * p^b * q^c
where a is 0 or 1, b is 0 or 1 or 2, and c is 0 or 1 or 2.
Going back to 22 as a specific example, the possible values of z are
1, 2, 4, -1, -2, -4,
11, 22, 44, -11, -22, -44,
121, 242, 484, -121, -242, -484
Arranging these in ascending order and translating back to x, we get:
(z -> x, y)
z <= -22 leads to x <= 0 (invalid)
-11 -> 11, -22
-4 -> 18, -99
-2 -> 20, -220
-1 -> 21, -462
+1 -> 23, 506
+2 -> 24, 264
+4 -> 26, 143
+11 -> 33, 66
+22 -> 44, 44
z > +22 leads to x > y (swap of one of the above, or invalid)
In general, I think the valid answers always end up being the
following (without loss of generality, let p < q):
y = pqx/(x - pq)
z = x - pq
(z < -pq leads to x < 0, invalid. -pq, -ppq, -qq, -pqq, and -ppqq
all fall into this category.)
-q -> q, -pq
-p^2
-p
-1
+1
+p
+p^2
+q
+pq -> 2pq, 2pq
(z > +pq is a swap of one of the others, or invalid. ppq, qq, pqq, and
ppqq all fall into this category.)
The only part of the ascending sequence that varies is whether p^2 or q
is larger (and thus whether -p^2 or -q is smaller). Either way, pq is
still smaller than either p^2 or q (and -pq is smaller than either -p^2
or -q).
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