The Mathematics Education Department
EMT 725, Lisa F. Garey

Sum of Unit Fractions

Click here for the Problem statement

Let x and y be the integers that are relatively prime. Then . However, there is a condition on the sum of these unit fractions. x+y+1 = xy. Solving these two equations,

. Substituting, which gives x=2. Hence y=3.

Therefore, the sum of the unit fractions 1/2 + 1/3 = 5/6 is in the form n/n+1.

Next, let's consider integers that are not re;atively prime. Let x and kx be the integers.

Then and k+1+1=k+2=kx since the sum must have the form n/n+1.

Solving these two equations,

Hence, k=2 and x=2.

Therefore, 1/2 + 1/4 = 3/4 is in the form n/n+1.

Further discussions with my colleagues revealed other cases that algebraic methods do not yield. Thus far, the sum of the unit fractions have been in its simplest form. We have not considered the case when the numerator and denominator in the sum of the unit fractions are not relatively prime, but reduces to one of the form n/n+1. In this case, a spreadsheet comes to the rescue.

1	2	3	4	5	6	7	8	9
2	####	5	3	2.33	1	1.8	1.667	1.571
3	5	2	1.4	1.142	0.692	0.909	0.846	0.8
4	3	1.4	1	0.818	0.556	0.647	0.6	0.565
5	2.33	1.14289	0.8181	0.66	0.478	0.521	0.481	0.451
6	2	1	0.714	0.578	0.428	0.448	0.411	0.384
7	1.8	0.909	0.647	0.521	0.393	0.4	0.365	0.340
8	1.667	0.8461	0.6	0.481	0.368	0.365	0.333	0.309
9	1.571	0.8	0.565	0.451	0.348	0.340	0.309	0.285
10	1.5	0.764	0.538	0.428	0.333	0.320	0.290	0.267
11	1.444	0.736	0.517	0.410	0.320	0.305	0.275
12	1.4	0.714	0.5	0.395	0.310	0.292	0.263
13	1.36	0.695	0.485	0.382	0.301	0.281	0.253
14	1.333	0.68	0.473	0.372	0.294	0.272	0.244
15	1.308	0.666	0.463	0.36	0.287	0.265	0.237

The entries in the table is the value n=(x+y)/(xy-x-y). We know that n must be an integer.

Note the entries with n=1 are not valid since they involve the sum of 2 equal unit fractions.

Thus, the unit fractions whose sums is in the form n/n+1 are

1/2 + 1/3 = 5/6

1/2 + 1/4 = 3/4

1/2 + 1/6 = 4/6 =2/3

1/3 + 1/6 = 3/6 = 1/2

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