Math Challenged

[hhEb09'1]

How many solutions does this Diophantine equation have?

1/x +1/y = 1/10^10

x=y=1/(2x10^10) is one solution. Looks like it may be the only one, but I'm not yet completely convinced.

[The_Radiation_Specialist]

Another is

1/312510000000000 + 1/10000320000

And there's lots more...

[hhEb09'1]

Now I'm definitely not convinced!

x=15*10^9, y=30*10^9
x=11*10^9, y=110*10^9

[tdvance]

It transforms easily to x+y=xy/10^10 is a quadratic equation in two
variables. We change variables to make it easier to work with:

Let a=10^10 to reduce typing.

ax + ay + xy = 0

let x' -> x+a

a(x'-a) + ay + (x'-a)y = 0
ax' - a^2 +x'y = 0

let y' -> y+a

ax' - a^2 + x'(y-a)=0

a^2 + x'y' = 0

The affine transformation is bijective on the integers, so finding
integer solutions to this simpler equation gives integer solutions to
the original one. We only need be careful to throw away anything
giving x=0 or y=0.

a^2 is 10^20 which factors as 2^20*5^20, and thus we have a total of
21*21=441 pairs (x',y') of integers giving x'y'=a^2. Then, we'd take
x=x'-a, y=y'-a. Thus, we throw out any pairs x',y' with either of
them equal to a. That is only one case: x'=y'=a. This leaves 440
solutions.

[tdvance]

wait--I flipped a sign--one more time:

[tdvance]

It transforms easily to x+y=xy/10^10 is a quadratic equation in two
variables. We change variables to make it easier to work with:

Let a=10^10 to reduce typing.

ax + ay - xy = 0

let x' -> x-a

a(x'+a) + ay - (x'+a)y = 0
ax' + a^2 - x'y = 0

let y' -> y - a

ax' + a^2 - x'(y + a)=0

a^2 - x'y' = 0

The affine transformation is bijective on the integers, so finding
integer solutions to this simpler equation gives integer solutions to
the original one. We only need be careful to throw away anything
giving x=0 or y=0.

a^2 is 10^20 which factors as 2^20*5^20, and thus we have a total of
21*21=441 pairs (x',y') of integers giving x'y'=a^2. Then, we'd take
x=x'-a, y=y'-a. Thus, we throw out any pairs x',y' with either of
them equal to a. That is only one case: x'=y'=a. This leaves 440
solutions.

[tdvance]

add one more--x'=0 and y'=0 gives x=-a, y=-a, which is a solution--it's symmetric--throw away one solution after the affine transformation, put an analogous solution back in the original problem.

so, 441 solutions. This time it's my final answer.

[The_Radiation_Specialist]

Hmmm, my own solution


I get
1/x +1/y = 1/n
n= (xy)/(x+y)

there are three cases of x and y.

1) They are coprime - this is not possible since n is an integer.
2) they have some prime factors in common but some not in common. i.e. x= p1p2p3 and y = p1p2p4

then (xy)/(x+y)
= (p1p2p3p1p2p4)/(p1p2p3+ p1p2p4)
= (p1p2p3p1p2p4)/(p1p2(p3+p4))
= p3p1p2p4/(p3+p4), which also cannot be an integer

3) They are of the form (Without loss of generality) x= qy - this is the only possible case.

Therefore we reduce the equation to

n= (qyy)/(qy+y)
= (qy)/(q+1)
since q is coprime to q+1, y must be of the form y = m(q+1)

and hence our original equationg reduces to
= (qy)/(q+1)
= qm(q+1)/(q+1)
= qm
or n= qm

the number of solutions for q and m are equivalent to the number of divisors of n. each solution of q and m corresponds to a new solution with x and y. Infact this is how I generated the example above.

10^10 = 2^10 *5^10 and so the number of all the divisors you can generate is = (1+10)*(1)10) = 121

[Sarawak]

add one more--x'=0 and y'=0 gives x=-a, y=-a, which is a solution--it's symmetric--throw away one solution after the affine transformation, put an analogous solution back in the original problem.

Are you sure this is what you meant to write?

[Sarawak]

Are solutions with x or y negative allowed? I think this accounts for some or all of the difference between tdvance's answer and [b]The Radiation Specialist[b]'s answer.

[tdvance]

I'd say yes to negative solutions, because it is a Diophantine equation. That might be all the difference there is between the two solutions--there are more than twice as many solutions involving negatives than involving only positives, since there are several cases with one positive, one negative. As for post #339--I definitely wrote that part poorly, it's like I was saying in another thread, where you have a concise mathematical concept in your head, but putting it into words becomes challenging.

[tdvance]

quick perl script to check yields the following 441 solutions:

(check that last value of each line is always 1, first two values integers and no dupes--note--I see no negatives! it's the +a in the reverse transformation that does that, I guess.)

x=1.0000000001e+20, y=10000000001 -> 10^10*(1/x+1/y)=1
x=10000000001, y=1.0000000001e+20 -> 10^10*(1/x+1/y)=1
x=10000000002, y=5.000000001e+19 -> 10^10*(1/x+1/y)=1
x=10000000004, y=2.500000001e+19 -> 10^10*(1/x+1/y)=1
x=10000000005, y=2.000000001e+19 -> 10^10*(1/x+1/y)=1
x=10000000008, y=12500000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000010, y=10000000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000010000000000, y=10000000010 -> 10^10*(1/x+1/y)=1
x=10000000016, y=6250000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000020, y=5000000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000025, y=4000000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000032, y=3125000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000040, y=2500000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000050, y=2000000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000064, y=1562500010000000000 -> 10^10*(1/x+1/y)=1
x=10000000080, y=1250000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000100, y=1000000010000000000 -> 10^10*(1/x+1/y)=1
x=1000000010000000000, y=10000000100 -> 10^10*(1/x+1/y)=1
x=10000000125, y=800000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000128, y=781250010000000000 -> 10^10*(1/x+1/y)=1
x=10000000160, y=625000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000200, y=500000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000250, y=400000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000256, y=390625010000000000 -> 10^10*(1/x+1/y)=1
x=10000000320, y=312500010000000000 -> 10^10*(1/x+1/y)=1
x=10000000400, y=250000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000500, y=200000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000512, y=195312510000000000 -> 10^10*(1/x+1/y)=1
x=10000000625, y=160000010000000000 -> 10^10*(1/x+1/y)=1
x=10000000640, y=156250010000000000 -> 10^10*(1/x+1/y)=1
x=10000000800, y=125000010000000000 -> 10^10*(1/x+1/y)=1
x=10000001000, y=100000010000000000 -> 10^10*(1/x+1/y)=1
x=100000010000000000, y=10000001000 -> 10^10*(1/x+1/y)=1
x=10000001024, y=97656260000000000 -> 10^10*(1/x+1/y)=1
x=10000001250, y=80000010000000000 -> 10^10*(1/x+1/y)=1
x=10000001280, y=78125010000000000 -> 10^10*(1/x+1/y)=1
x=10000001600, y=62500010000000000 -> 10^10*(1/x+1/y)=1
x=10000002000, y=50000010000000000 -> 10^10*(1/x+1/y)=1
x=10000002048, y=48828135000000000 -> 10^10*(1/x+1/y)=1
x=10000002500, y=40000010000000000 -> 10^10*(1/x+1/y)=1
x=10000002560, y=39062510000000000 -> 10^10*(1/x+1/y)=1
x=10000003125, y=32000010000000000 -> 10^10*(1/x+1/y)=1
x=10000003200, y=31250010000000000 -> 10^10*(1/x+1/y)=1
x=10000004000, y=25000010000000000 -> 10^10*(1/x+1/y)=1
x=10000004096, y=24414072500000000 -> 10^10*(1/x+1/y)=1
x=10000005000, y=20000010000000000 -> 10^10*(1/x+1/y)=1
x=10000005120, y=19531260000000000 -> 10^10*(1/x+1/y)=1
x=10000006250, y=16000010000000000 -> 10^10*(1/x+1/y)=1
x=10000006400, y=15625010000000000 -> 10^10*(1/x+1/y)=1
x=10000008000, y=12500010000000000 -> 10^10*(1/x+1/y)=1
x=10000008192, y=12207041250000000 -> 10^10*(1/x+1/y)=1
x=10000010000, y=10000010000000000 -> 10^10*(1/x+1/y)=1
x=10000010000000000, y=10000010000 -> 10^10*(1/x+1/y)=1
x=10000010240, y=9765635000000000 -> 10^10*(1/x+1/y)=1
x=10000012500, y=8000010000000000 -> 10^10*(1/x+1/y)=1
x=10000012800, y=7812510000000000 -> 10^10*(1/x+1/y)=1
x=10000015625, y=6400010000000000 -> 10^10*(1/x+1/y)=1
x=10000016000, y=6250010000000000 -> 10^10*(1/x+1/y)=1
x=10000016384, y=6103525625000000 -> 10^10*(1/x+1/y)=1
x=10000020000, y=5000010000000000 -> 10^10*(1/x+1/y)=1
x=10000020480, y=4882822500000000 -> 10^10*(1/x+1/y)=1
x=10000025000, y=4000010000000000 -> 10^10*(1/x+1/y)=1
x=10000025600, y=3906260000000000 -> 10^10*(1/x+1/y)=1
x=10000031250, y=3200010000000000 -> 10^10*(1/x+1/y)=1
x=10000032000, y=3125010000000000 -> 10^10*(1/x+1/y)=1
x=10000032768, y=3051767812500000 -> 10^10*(1/x+1/y)=1
x=10000040000, y=2500010000000000 -> 10^10*(1/x+1/y)=1
x=10000040960, y=2441416250000000 -> 10^10*(1/x+1/y)=1
x=10000050000, y=2000010000000000 -> 10^10*(1/x+1/y)=1
x=10000051200, y=1953135000000000 -> 10^10*(1/x+1/y)=1
x=10000062500, y=1600010000000000 -> 10^10*(1/x+1/y)=1
x=10000064000, y=1562510000000000 -> 10^10*(1/x+1/y)=1
x=10000065536, y=1525888906250000 -> 10^10*(1/x+1/y)=1
x=10000078125, y=1280010000000000 -> 10^10*(1/x+1/y)=1
x=10000080000, y=1250010000000000 -> 10^10*(1/x+1/y)=1
x=10000081920, y=1220713125000000 -> 10^10*(1/x+1/y)=1
x=10000100000, y=1000010000000000 -> 10^10*(1/x+1/y)=1
x=1000010000000000, y=10000100000 -> 10^10*(1/x+1/y)=1
x=10000102400, y=976572500000000 -> 10^10*(1/x+1/y)=1
x=10000125000, y=800010000000000 -> 10^10*(1/x+1/y)=1
x=10000128000, y=781260000000000 -> 10^10*(1/x+1/y)=1
x=10000131072, y=762949453125000 -> 10^10*(1/x+1/y)=1
x=10000156250, y=640010000000000 -> 10^10*(1/x+1/y)=1
x=10000160000, y=625010000000000 -> 10^10*(1/x+1/y)=1
x=10000163840, y=610361562500000 -> 10^10*(1/x+1/y)=1
x=10000200000, y=500010000000000 -> 10^10*(1/x+1/y)=1
x=10000204800, y=488291250000000 -> 10^10*(1/x+1/y)=1
x=10000250000, y=400010000000000 -> 10^10*(1/x+1/y)=1
x=10000256000, y=390635000000000 -> 10^10*(1/x+1/y)=1
x=10000262144, y=381479726562500 -> 10^10*(1/x+1/y)=1
x=10000312500, y=320010000000000 -> 10^10*(1/x+1/y)=1
x=10000320000, y=312510000000000 -> 10^10*(1/x+1/y)=1
x=10000327680, y=305185781250000 -> 10^10*(1/x+1/y)=1
x=10000390625, y=256010000000000 -> 10^10*(1/x+1/y)=1
x=10000400000, y=250010000000000 -> 10^10*(1/x+1/y)=1
x=10000409600, y=244150625000000 -> 10^10*(1/x+1/y)=1
x=10000500000, y=200010000000000 -> 10^10*(1/x+1/y)=1
x=10000512000, y=195322500000000 -> 10^10*(1/x+1/y)=1
x=10000524288, y=190744863281250 -> 10^10*(1/x+1/y)=1
x=10000625000, y=160010000000000 -> 10^10*(1/x+1/y)=1
x=10000640000, y=156260000000000 -> 10^10*(1/x+1/y)=1
x=10000655360, y=152597890625000 -> 10^10*(1/x+1/y)=1
x=10000781250, y=128010000000000 -> 10^10*(1/x+1/y)=1
x=10000800000, y=125010000000000 -> 10^10*(1/x+1/y)=1
x=10000819200, y=122080312500000 -> 10^10*(1/x+1/y)=1
x=10001000000, y=100010000000000 -> 10^10*(1/x+1/y)=1
x=100010000000000, y=10001000000 -> 10^10*(1/x+1/y)=1
x=10001024000, y=97666250000000 -> 10^10*(1/x+1/y)=1
x=10001048576, y=95377431640625 -> 10^10*(1/x+1/y)=1
x=10001250000, y=80010000000000 -> 10^10*(1/x+1/y)=1
x=10001280000, y=78135000000000 -> 10^10*(1/x+1/y)=1
x=10001310720, y=76303945312500 -> 10^10*(1/x+1/y)=1
x=10001562500, y=64010000000000 -> 10^10*(1/x+1/y)=1
x=10001600000, y=62510000000000 -> 10^10*(1/x+1/y)=1
x=10001638400, y=61045156250000 -> 10^10*(1/x+1/y)=1
x=10001953125, y=51210000000000 -> 10^10*(1/x+1/y)=1
x=10002000000, y=50010000000000 -> 10^10*(1/x+1/y)=1
x=10002048000, y=48838125000000 -> 10^10*(1/x+1/y)=1
x=10002500000, y=40010000000000 -> 10^10*(1/x+1/y)=1
x=10002560000, y=39072500000000 -> 10^10*(1/x+1/y)=1
x=10002621440, y=38156972656250 -> 10^10*(1/x+1/y)=1
x=10003125000, y=32010000000000 -> 10^10*(1/x+1/y)=1
x=10003200000, y=31260000000000 -> 10^10*(1/x+1/y)=1
x=10003276800, y=30527578125000 -> 10^10*(1/x+1/y)=1
x=10003906250, y=25610000000000 -> 10^10*(1/x+1/y)=1
x=10004000000, y=25010000000000 -> 10^10*(1/x+1/y)=1
x=10004096000, y=24424062500000 -> 10^10*(1/x+1/y)=1
x=10005000000, y=20010000000000 -> 10^10*(1/x+1/y)=1
x=10005120000, y=19541250000000 -> 10^10*(1/x+1/y)=1
x=10005242880, y=19083486328125 -> 10^10*(1/x+1/y)=1
x=10006250000, y=16010000000000 -> 10^10*(1/x+1/y)=1
x=10006400000, y=15635000000000 -> 10^10*(1/x+1/y)=1
x=10006553600, y=15268789062500 -> 10^10*(1/x+1/y)=1
x=10007812500, y=12810000000000 -> 10^10*(1/x+1/y)=1
x=10008000000, y=12510000000000 -> 10^10*(1/x+1/y)=1
x=10008192000, y=12217031250000 -> 10^10*(1/x+1/y)=1
x=10009765625, y=10250000000000 -> 10^10*(1/x+1/y)=1
x=10010000000, y=10010000000000 -> 10^10*(1/x+1/y)=1
x=10010000000000, y=10010000000 -> 10^10*(1/x+1/y)=1
x=10010240000, y=9775625000000 -> 10^10*(1/x+1/y)=1
x=10012500000, y=8010000000000 -> 10^10*(1/x+1/y)=1
x=10012800000, y=7822500000000 -> 10^10*(1/x+1/y)=1
x=10013107200, y=7639394531250 -> 10^10*(1/x+1/y)=1
x=10015625000, y=6410000000000 -> 10^10*(1/x+1/y)=1
x=10016000000, y=6260000000000 -> 10^10*(1/x+1/y)=1
x=10016384000, y=6113515625000 -> 10^10*(1/x+1/y)=1
x=10019531250, y=5130000000000 -> 10^10*(1/x+1/y)=1
x=10020000000, y=5010000000000 -> 10^10*(1/x+1/y)=1
x=10020480000, y=4892812500000 -> 10^10*(1/x+1/y)=1
x=10025000000, y=4010000000000 -> 10^10*(1/x+1/y)=1
x=10025600000, y=3916250000000 -> 10^10*(1/x+1/y)=1
x=10026214400, y=3824697265625 -> 10^10*(1/x+1/y)=1
x=10031250000, y=3210000000000 -> 10^10*(1/x+1/y)=1
x=10032000000, y=3135000000000 -> 10^10*(1/x+1/y)=1
x=10032768000, y=3061757812500 -> 10^10*(1/x+1/y)=1
x=10039062500, y=2570000000000 -> 10^10*(1/x+1/y)=1
x=10040000000, y=2510000000000 -> 10^10*(1/x+1/y)=1
x=10040960000, y=2451406250000 -> 10^10*(1/x+1/y)=1
x=10048828125, y=2058000000000 -> 10^10*(1/x+1/y)=1
x=10050000000, y=2010000000000 -> 10^10*(1/x+1/y)=1
x=10051200000, y=1963125000000 -> 10^10*(1/x+1/y)=1
x=10062500000, y=1610000000000 -> 10^10*(1/x+1/y)=1
x=10064000000, y=1572500000000 -> 10^10*(1/x+1/y)=1
x=10065536000, y=1535878906250 -> 10^10*(1/x+1/y)=1
x=10078125000, y=1290000000000 -> 10^10*(1/x+1/y)=1
x=10080000000, y=1260000000000 -> 10^10*(1/x+1/y)=1
x=10081920000, y=1230703125000 -> 10^10*(1/x+1/y)=1
x=10097656250, y=1034000000000 -> 10^10*(1/x+1/y)=1
x=10100000000, y=1010000000000 -> 10^10*(1/x+1/y)=1
x=1010000000000, y=10100000000 -> 10^10*(1/x+1/y)=1
x=10102400000, y=986562500000 -> 10^10*(1/x+1/y)=1
x=10125000000, y=810000000000 -> 10^10*(1/x+1/y)=1
x=10128000000, y=791250000000 -> 10^10*(1/x+1/y)=1
x=10131072000, y=772939453125 -> 10^10*(1/x+1/y)=1
x=10156250000, y=650000000000 -> 10^10*(1/x+1/y)=1
x=10160000000, y=635000000000 -> 10^10*(1/x+1/y)=1
x=10163840000, y=620351562500 -> 10^10*(1/x+1/y)=1
x=10195312500, y=522000000000 -> 10^10*(1/x+1/y)=1
x=10200000000, y=510000000000 -> 10^10*(1/x+1/y)=1
x=10204800000, y=498281250000 -> 10^10*(1/x+1/y)=1
x=10244140625, y=419600000000 -> 10^10*(1/x+1/y)=1
x=10250000000, y=410000000000 -> 10^10*(1/x+1/y)=1
x=10250000000000, y=10009765625 -> 10^10*(1/x+1/y)=1
x=10256000000, y=400625000000 -> 10^10*(1/x+1/y)=1
x=10312500000, y=330000000000 -> 10^10*(1/x+1/y)=1
x=10320000000, y=322500000000 -> 10^10*(1/x+1/y)=1
x=10327680000, y=315175781250 -> 10^10*(1/x+1/y)=1
x=1034000000000, y=10097656250 -> 10^10*(1/x+1/y)=1
x=10390625000, y=266000000000 -> 10^10*(1/x+1/y)=1
x=10400000000, y=260000000000 -> 10^10*(1/x+1/y)=1
x=10409600000, y=254140625000 -> 10^10*(1/x+1/y)=1
x=10488281250, y=214800000000 -> 10^10*(1/x+1/y)=1
x=10500000000, y=210000000000 -> 10^10*(1/x+1/y)=1
x=10512000000, y=205312500000 -> 10^10*(1/x+1/y)=1
x=10625000000, y=170000000000 -> 10^10*(1/x+1/y)=1
x=10640000000, y=166250000000 -> 10^10*(1/x+1/y)=1
x=10655360000, y=162587890625 -> 10^10*(1/x+1/y)=1

[tdvance]

2nd half:

x=107656250000, y=11024000000 -> 10^10*(1/x+1/y)=1
x=10781250000, y=138000000000 -> 10^10*(1/x+1/y)=1
x=10800000000, y=135000000000 -> 10^10*(1/x+1/y)=1
x=10819200000, y=132070312500 -> 10^10*(1/x+1/y)=1
x=10976562500, y=112400000000 -> 10^10*(1/x+1/y)=1
x=11000000000, y=110000000000 -> 10^10*(1/x+1/y)=1
x=110000000000, y=11000000000 -> 10^10*(1/x+1/y)=1
x=11024000000, y=107656250000 -> 10^10*(1/x+1/y)=1
x=11220703125, y=91920000000 -> 10^10*(1/x+1/y)=1
x=112400000000, y=10976562500 -> 10^10*(1/x+1/y)=1
x=11250000000, y=90000000000 -> 10^10*(1/x+1/y)=1
x=11280000000, y=88125000000 -> 10^10*(1/x+1/y)=1
x=11562500000, y=74000000000 -> 10^10*(1/x+1/y)=1
x=11600000000, y=72500000000 -> 10^10*(1/x+1/y)=1
x=11638400000, y=71035156250 -> 10^10*(1/x+1/y)=1
x=11953125000, y=61200000000 -> 10^10*(1/x+1/y)=1
x=12000000000, y=60000000000 -> 10^10*(1/x+1/y)=1
x=12048000000, y=58828125000 -> 10^10*(1/x+1/y)=1
x=12207041250000000, y=10000008192 -> 10^10*(1/x+1/y)=1
x=1220713125000000, y=10000081920 -> 10^10*(1/x+1/y)=1
x=122080312500000, y=10000819200 -> 10^10*(1/x+1/y)=1
x=12217031250000, y=10008192000 -> 10^10*(1/x+1/y)=1
x=1230703125000, y=10081920000 -> 10^10*(1/x+1/y)=1
x=12441406250, y=50960000000 -> 10^10*(1/x+1/y)=1
x=12500000000, y=50000000000 -> 10^10*(1/x+1/y)=1
x=12500000010000000000, y=10000000008 -> 10^10*(1/x+1/y)=1
x=1250000010000000000, y=10000000080 -> 10^10*(1/x+1/y)=1
x=125000010000000000, y=10000000800 -> 10^10*(1/x+1/y)=1
x=12500010000000000, y=10000008000 -> 10^10*(1/x+1/y)=1
x=1250010000000000, y=10000080000 -> 10^10*(1/x+1/y)=1
x=125010000000000, y=10000800000 -> 10^10*(1/x+1/y)=1
x=12510000000000, y=10008000000 -> 10^10*(1/x+1/y)=1
x=12560000000, y=49062500000 -> 10^10*(1/x+1/y)=1
x=1260000000000, y=10080000000 -> 10^10*(1/x+1/y)=1
x=1280010000000000, y=10000078125 -> 10^10*(1/x+1/y)=1
x=128010000000000, y=10000781250 -> 10^10*(1/x+1/y)=1
x=12810000000000, y=10007812500 -> 10^10*(1/x+1/y)=1
x=1290000000000, y=10078125000 -> 10^10*(1/x+1/y)=1
x=13125000000, y=42000000000 -> 10^10*(1/x+1/y)=1
x=13200000000, y=41250000000 -> 10^10*(1/x+1/y)=1
x=132070312500, y=10819200000 -> 10^10*(1/x+1/y)=1
x=13276800000, y=40517578125 -> 10^10*(1/x+1/y)=1
x=135000000000, y=10800000000 -> 10^10*(1/x+1/y)=1
x=138000000000, y=10781250000 -> 10^10*(1/x+1/y)=1
x=13906250000, y=35600000000 -> 10^10*(1/x+1/y)=1
x=14000000000, y=35000000000 -> 10^10*(1/x+1/y)=1
x=14096000000, y=34414062500 -> 10^10*(1/x+1/y)=1
x=14882812500, y=30480000000 -> 10^10*(1/x+1/y)=1
x=15000000000, y=30000000000 -> 10^10*(1/x+1/y)=1
x=15120000000, y=29531250000 -> 10^10*(1/x+1/y)=1
x=1525888906250000, y=10000065536 -> 10^10*(1/x+1/y)=1
x=152597890625000, y=10000655360 -> 10^10*(1/x+1/y)=1
x=15268789062500, y=10006553600 -> 10^10*(1/x+1/y)=1
x=1535878906250, y=10065536000 -> 10^10*(1/x+1/y)=1
x=1562500010000000000, y=10000000064 -> 10^10*(1/x+1/y)=1
x=156250010000000000, y=10000000640 -> 10^10*(1/x+1/y)=1
x=15625010000000000, y=10000006400 -> 10^10*(1/x+1/y)=1
x=1562510000000000, y=10000064000 -> 10^10*(1/x+1/y)=1
x=156260000000000, y=10000640000 -> 10^10*(1/x+1/y)=1
x=15635000000000, y=10006400000 -> 10^10*(1/x+1/y)=1
x=1572500000000, y=10064000000 -> 10^10*(1/x+1/y)=1
x=160000010000000000, y=10000000625 -> 10^10*(1/x+1/y)=1
x=16000010000000000, y=10000006250 -> 10^10*(1/x+1/y)=1
x=1600010000000000, y=10000062500 -> 10^10*(1/x+1/y)=1
x=160010000000000, y=10000625000 -> 10^10*(1/x+1/y)=1
x=16010000000000, y=10006250000 -> 10^10*(1/x+1/y)=1
x=1610000000000, y=10062500000 -> 10^10*(1/x+1/y)=1
x=16103515625, y=26384000000 -> 10^10*(1/x+1/y)=1
x=16250000000, y=26000000000 -> 10^10*(1/x+1/y)=1
x=162587890625, y=10655360000 -> 10^10*(1/x+1/y)=1
x=16400000000, y=25625000000 -> 10^10*(1/x+1/y)=1
x=166250000000, y=10640000000 -> 10^10*(1/x+1/y)=1
x=170000000000, y=10625000000 -> 10^10*(1/x+1/y)=1
x=17812500000, y=22800000000 -> 10^10*(1/x+1/y)=1
x=18000000000, y=22500000000 -> 10^10*(1/x+1/y)=1
x=18192000000, y=22207031250 -> 10^10*(1/x+1/y)=1
x=190744863281250, y=10000524288 -> 10^10*(1/x+1/y)=1
x=19083486328125, y=10005242880 -> 10^10*(1/x+1/y)=1
x=195312510000000000, y=10000000512 -> 10^10*(1/x+1/y)=1
x=19531260000000000, y=10000005120 -> 10^10*(1/x+1/y)=1
x=1953135000000000, y=10000051200 -> 10^10*(1/x+1/y)=1
x=195322500000000, y=10000512000 -> 10^10*(1/x+1/y)=1
x=19541250000000, y=10005120000 -> 10^10*(1/x+1/y)=1
x=1963125000000, y=10051200000 -> 10^10*(1/x+1/y)=1
x=19765625000, y=20240000000 -> 10^10*(1/x+1/y)=1
x=2.000000001e+19, y=10000000005 -> 10^10*(1/x+1/y)=1
x=2.500000001e+19, y=10000000004 -> 10^10*(1/x+1/y)=1
x=20000000000, y=20000000000 -> 10^10*(1/x+1/y)=1
x=2000000010000000000, y=10000000050 -> 10^10*(1/x+1/y)=1
x=200000010000000000, y=10000000500 -> 10^10*(1/x+1/y)=1
x=20000010000000000, y=10000005000 -> 10^10*(1/x+1/y)=1
x=2000010000000000, y=10000050000 -> 10^10*(1/x+1/y)=1
x=200010000000000, y=10000500000 -> 10^10*(1/x+1/y)=1
x=20010000000000, y=10005000000 -> 10^10*(1/x+1/y)=1
x=2010000000000, y=10050000000 -> 10^10*(1/x+1/y)=1
x=20240000000, y=19765625000 -> 10^10*(1/x+1/y)=1
x=205312500000, y=10512000000 -> 10^10*(1/x+1/y)=1
x=2058000000000, y=10048828125 -> 10^10*(1/x+1/y)=1
x=210000000000, y=10500000000 -> 10^10*(1/x+1/y)=1
x=214800000000, y=10488281250 -> 10^10*(1/x+1/y)=1
x=22207031250, y=18192000000 -> 10^10*(1/x+1/y)=1
x=22500000000, y=18000000000 -> 10^10*(1/x+1/y)=1
x=22800000000, y=17812500000 -> 10^10*(1/x+1/y)=1
x=24414072500000000, y=10000004096 -> 10^10*(1/x+1/y)=1
x=2441416250000000, y=10000040960 -> 10^10*(1/x+1/y)=1
x=244150625000000, y=10000409600 -> 10^10*(1/x+1/y)=1
x=24424062500000, y=10004096000 -> 10^10*(1/x+1/y)=1
x=2451406250000, y=10040960000 -> 10^10*(1/x+1/y)=1
x=2500000010000000000, y=10000000040 -> 10^10*(1/x+1/y)=1
x=250000010000000000, y=10000000400 -> 10^10*(1/x+1/y)=1
x=25000010000000000, y=10000004000 -> 10^10*(1/x+1/y)=1
x=2500010000000000, y=10000040000 -> 10^10*(1/x+1/y)=1
x=250010000000000, y=10000400000 -> 10^10*(1/x+1/y)=1
x=25010000000000, y=10004000000 -> 10^10*(1/x+1/y)=1
x=2510000000000, y=10040000000 -> 10^10*(1/x+1/y)=1
x=254140625000, y=10409600000 -> 10^10*(1/x+1/y)=1
x=256010000000000, y=10000390625 -> 10^10*(1/x+1/y)=1
x=25610000000000, y=10003906250 -> 10^10*(1/x+1/y)=1
x=25625000000, y=16400000000 -> 10^10*(1/x+1/y)=1
x=2570000000000, y=10039062500 -> 10^10*(1/x+1/y)=1
x=26000000000, y=16250000000 -> 10^10*(1/x+1/y)=1
x=260000000000, y=10400000000 -> 10^10*(1/x+1/y)=1
x=26384000000, y=16103515625 -> 10^10*(1/x+1/y)=1
x=266000000000, y=10390625000 -> 10^10*(1/x+1/y)=1
x=29531250000, y=15120000000 -> 10^10*(1/x+1/y)=1
x=30000000000, y=15000000000 -> 10^10*(1/x+1/y)=1
x=30480000000, y=14882812500 -> 10^10*(1/x+1/y)=1
x=3051767812500000, y=10000032768 -> 10^10*(1/x+1/y)=1
x=305185781250000, y=10000327680 -> 10^10*(1/x+1/y)=1
x=30527578125000, y=10003276800 -> 10^10*(1/x+1/y)=1
x=3061757812500, y=10032768000 -> 10^10*(1/x+1/y)=1
x=3125000010000000000, y=10000000032 -> 10^10*(1/x+1/y)=1
x=312500010000000000, y=10000000320 -> 10^10*(1/x+1/y)=1
x=31250010000000000, y=10000003200 -> 10^10*(1/x+1/y)=1
x=3125010000000000, y=10000032000 -> 10^10*(1/x+1/y)=1
x=312510000000000, y=10000320000 -> 10^10*(1/x+1/y)=1
x=31260000000000, y=10003200000 -> 10^10*(1/x+1/y)=1
x=3135000000000, y=10032000000 -> 10^10*(1/x+1/y)=1
x=315175781250, y=10327680000 -> 10^10*(1/x+1/y)=1
x=32000010000000000, y=10000003125 -> 10^10*(1/x+1/y)=1
x=3200010000000000, y=10000031250 -> 10^10*(1/x+1/y)=1
x=320010000000000, y=10000312500 -> 10^10*(1/x+1/y)=1
x=32010000000000, y=10003125000 -> 10^10*(1/x+1/y)=1
x=3210000000000, y=10031250000 -> 10^10*(1/x+1/y)=1
x=322500000000, y=10320000000 -> 10^10*(1/x+1/y)=1
x=330000000000, y=10312500000 -> 10^10*(1/x+1/y)=1
x=34414062500, y=14096000000 -> 10^10*(1/x+1/y)=1
x=35000000000, y=14000000000 -> 10^10*(1/x+1/y)=1
x=35600000000, y=13906250000 -> 10^10*(1/x+1/y)=1
x=381479726562500, y=10000262144 -> 10^10*(1/x+1/y)=1
x=38156972656250, y=10002621440 -> 10^10*(1/x+1/y)=1
x=3824697265625, y=10026214400 -> 10^10*(1/x+1/y)=1
x=390625010000000000, y=10000000256 -> 10^10*(1/x+1/y)=1
x=39062510000000000, y=10000002560 -> 10^10*(1/x+1/y)=1
x=3906260000000000, y=10000025600 -> 10^10*(1/x+1/y)=1
x=390635000000000, y=10000256000 -> 10^10*(1/x+1/y)=1
x=39072500000000, y=10002560000 -> 10^10*(1/x+1/y)=1
x=3916250000000, y=10025600000 -> 10^10*(1/x+1/y)=1
x=4000000010000000000, y=10000000025 -> 10^10*(1/x+1/y)=1
x=400000010000000000, y=10000000250 -> 10^10*(1/x+1/y)=1
x=40000010000000000, y=10000002500 -> 10^10*(1/x+1/y)=1
x=4000010000000000, y=10000025000 -> 10^10*(1/x+1/y)=1
x=400010000000000, y=10000250000 -> 10^10*(1/x+1/y)=1
x=40010000000000, y=10002500000 -> 10^10*(1/x+1/y)=1
x=400625000000, y=10256000000 -> 10^10*(1/x+1/y)=1
x=4010000000000, y=10025000000 -> 10^10*(1/x+1/y)=1
x=40517578125, y=13276800000 -> 10^10*(1/x+1/y)=1
x=410000000000, y=10250000000 -> 10^10*(1/x+1/y)=1
x=41250000000, y=13200000000 -> 10^10*(1/x+1/y)=1
x=419600000000, y=10244140625 -> 10^10*(1/x+1/y)=1
x=42000000000, y=13125000000 -> 10^10*(1/x+1/y)=1
x=48828135000000000, y=10000002048 -> 10^10*(1/x+1/y)=1
x=4882822500000000, y=10000020480 -> 10^10*(1/x+1/y)=1
x=488291250000000, y=10000204800 -> 10^10*(1/x+1/y)=1
x=48838125000000, y=10002048000 -> 10^10*(1/x+1/y)=1
x=4892812500000, y=10020480000 -> 10^10*(1/x+1/y)=1
x=49062500000, y=12560000000 -> 10^10*(1/x+1/y)=1
x=498281250000, y=10204800000 -> 10^10*(1/x+1/y)=1
x=5.000000001e+19, y=10000000002 -> 10^10*(1/x+1/y)=1
x=50000000000, y=12500000000 -> 10^10*(1/x+1/y)=1
x=5000000010000000000, y=10000000020 -> 10^10*(1/x+1/y)=1
x=500000010000000000, y=10000000200 -> 10^10*(1/x+1/y)=1
x=50000010000000000, y=10000002000 -> 10^10*(1/x+1/y)=1
x=5000010000000000, y=10000020000 -> 10^10*(1/x+1/y)=1
x=500010000000000, y=10000200000 -> 10^10*(1/x+1/y)=1
x=50010000000000, y=10002000000 -> 10^10*(1/x+1/y)=1
x=5010000000000, y=10020000000 -> 10^10*(1/x+1/y)=1
x=50960000000, y=12441406250 -> 10^10*(1/x+1/y)=1
x=510000000000, y=10200000000 -> 10^10*(1/x+1/y)=1
x=51210000000000, y=10001953125 -> 10^10*(1/x+1/y)=1
x=5130000000000, y=10019531250 -> 10^10*(1/x+1/y)=1
x=522000000000, y=10195312500 -> 10^10*(1/x+1/y)=1
x=58828125000, y=12048000000 -> 10^10*(1/x+1/y)=1
x=60000000000, y=12000000000 -> 10^10*(1/x+1/y)=1
x=6103525625000000, y=10000016384 -> 10^10*(1/x+1/y)=1
x=610361562500000, y=10000163840 -> 10^10*(1/x+1/y)=1
x=61045156250000, y=10001638400 -> 10^10*(1/x+1/y)=1
x=6113515625000, y=10016384000 -> 10^10*(1/x+1/y)=1
x=61200000000, y=11953125000 -> 10^10*(1/x+1/y)=1
x=620351562500, y=10163840000 -> 10^10*(1/x+1/y)=1
x=6250000010000000000, y=10000000016 -> 10^10*(1/x+1/y)=1
x=625000010000000000, y=10000000160 -> 10^10*(1/x+1/y)=1
x=62500010000000000, y=10000001600 -> 10^10*(1/x+1/y)=1
x=6250010000000000, y=10000016000 -> 10^10*(1/x+1/y)=1
x=625010000000000, y=10000160000 -> 10^10*(1/x+1/y)=1
x=62510000000000, y=10001600000 -> 10^10*(1/x+1/y)=1
x=6260000000000, y=10016000000 -> 10^10*(1/x+1/y)=1
x=635000000000, y=10160000000 -> 10^10*(1/x+1/y)=1
x=6400010000000000, y=10000015625 -> 10^10*(1/x+1/y)=1
x=640010000000000, y=10000156250 -> 10^10*(1/x+1/y)=1
x=64010000000000, y=10001562500 -> 10^10*(1/x+1/y)=1
x=6410000000000, y=10015625000 -> 10^10*(1/x+1/y)=1
x=650000000000, y=10156250000 -> 10^10*(1/x+1/y)=1
x=71035156250, y=11638400000 -> 10^10*(1/x+1/y)=1
x=72500000000, y=11600000000 -> 10^10*(1/x+1/y)=1
x=74000000000, y=11562500000 -> 10^10*(1/x+1/y)=1
x=762949453125000, y=10000131072 -> 10^10*(1/x+1/y)=1
x=76303945312500, y=10001310720 -> 10^10*(1/x+1/y)=1
x=7639394531250, y=10013107200 -> 10^10*(1/x+1/y)=1
x=772939453125, y=10131072000 -> 10^10*(1/x+1/y)=1
x=781250010000000000, y=10000000128 -> 10^10*(1/x+1/y)=1
x=78125010000000000, y=10000001280 -> 10^10*(1/x+1/y)=1
x=7812510000000000, y=10000012800 -> 10^10*(1/x+1/y)=1
x=781260000000000, y=10000128000 -> 10^10*(1/x+1/y)=1
x=78135000000000, y=10001280000 -> 10^10*(1/x+1/y)=1
x=7822500000000, y=10012800000 -> 10^10*(1/x+1/y)=1
x=791250000000, y=10128000000 -> 10^10*(1/x+1/y)=1
x=800000010000000000, y=10000000125 -> 10^10*(1/x+1/y)=1
x=80000010000000000, y=10000001250 -> 10^10*(1/x+1/y)=1
x=8000010000000000, y=10000012500 -> 10^10*(1/x+1/y)=1
x=800010000000000, y=10000125000 -> 10^10*(1/x+1/y)=1
x=80010000000000, y=10001250000 -> 10^10*(1/x+1/y)=1
x=8010000000000, y=10012500000 -> 10^10*(1/x+1/y)=1
x=810000000000, y=10125000000 -> 10^10*(1/x+1/y)=1
x=88125000000, y=11280000000 -> 10^10*(1/x+1/y)=1
x=90000000000, y=11250000000 -> 10^10*(1/x+1/y)=1
x=91920000000, y=11220703125 -> 10^10*(1/x+1/y)=1
x=95377431640625, y=10001048576 -> 10^10*(1/x+1/y)=1
x=97656260000000000, y=10000001024 -> 10^10*(1/x+1/y)=1
x=9765635000000000, y=10000010240 -> 10^10*(1/x+1/y)=1
x=976572500000000, y=10000102400 -> 10^10*(1/x+1/y)=1
x=97666250000000, y=10001024000 -> 10^10*(1/x+1/y)=1
x=9775625000000, y=10010240000 -> 10^10*(1/x+1/y)=1
x=986562500000, y=10102400000 -> 10^10*(1/x+1/y)=1

[tdvance]

Are you sure this is what you meant to write?

should definitely be y=a, x=a--I was doing the wrong inverse inverse inverse inverse affine transformation.

[tdvance]

actually, I now realize, there are even more solutions if you use negatives. I tacitly assumed positive when I solved the transformed equation.

[The_Radiation_Specialist]

D'oh! there's a bug in my proof somewhere.

Since there's concrete evidence that the answer is right, your turn is next, sir.

[tdvance]

If integer n has 60 positive integer divisors, compute the largest number of positive integer divisors that n^2 could have.

[hhEb09'1]

It looks like 405 to me.

[tdvance]

Care to provide the justification (listing sample exponents would do, at least to dare someone to do better)?

[tdvance]

405 is the right answer, BTW.

[hhEb09'1]

I've not been busy over at the challenge

n=p¹q¹r²s⁴ is one way to get 60 divisors (2 x 2 x 3 x 5), with n² having (3 x 3 x 5 x 9) or 405.

m=p¹¹q⁴ aslo has 60 (12 x 5) but m² only has (23 x 9) which is 207.

[tdvance]

Correct answer--it is now hhEb09'1's turn.

[The_Radiation_Specialist]

an easy one just to get this thread back on track,

A wealthy man owned seventeen horses. When he died he bequeathed the horses to his three sons. The will stated that eldest son was to be given one half of the horses, the middle son was to be given one third of the horses, and the youngest son was to be given one ninth of the horses. The sons were distraught. It was clear to all that the horses could not divided in this way without making a bloody mess.

How should they go about?

[Sarawak]

They should have Pedro the Invisible Magic Horse join the other seventeen, and then divide them up. When they are finished, Pedro can go home.

[hhEb09'1]

Thanks for jumping in there to help me out, TRS! I got distracted, I guess.

They should have Pedro the Invisible Magic Horse join the other seventeen, and then divide them up. When they are finished, Pedro can go home.

The classic solution. The will only addressed dividing up one half plus one third plus one ninth of the horses, which gives each heir a partial horse, but 1/2 + 1/3 + 1/9 is only 17/18, which leaves 1/18 of the equine estate left to be parcelled out--and that can be done to give each of the three heirs their fair share, plus a little extra to make their share whole horses.

The young wicked stepmother, who was to get the rest of the estate, may disagree. They may have to compensate her somehow.

[Sarawak]

Thanks for jumping in there to help me out, TRS! I got distracted, I guess.

If you have a good question, I would say go ahead. It was your turn anyway, this was a time-filler question, and The Radiation Specialist is trying to get banned anyway.

[Susannah Dingley]

So whose turn is it now?

[Sarawak]

I just answered a question, but it is okay with me if anyone who has a good question, asks it. But I think hhEb09'1 has some claim to be next, so maybe it is best to check with him if you have a question

[hhEb09'1]

I just answered a question, but it is okay with me if anyone who has a good question, asks it. But I think hhEb09'1 has some claim to be next, so maybe it is best to check with him if you have a question

I would have said it is your turn, since TRS filled in for me. If you don't post a question within 24 hours, then I say open it up to anyone. I mean, that seems like a good policy in general, what do you all say? After 24 hours from the confirmation of a correct response, the floor is open--but only one (official) open question at a time?

[The_Radiation_Specialist]

Dang, this thread is among the few reasons I visit this forum and already it needs CPR every few days . . .

One from Project Euler...

A 30×30 grid of squares contains 900 fleas, initially one flea per square.
When a bell is rung, each flea jumps to an adjacent square at random (usually 4 possibilities, except for fleas on the edge of the grid or at the corners).

What is the expected number of unoccupied squares after 50 rings of the bell? Give your answer rounded to six decimal places.