A unit portion is a portion with a molecule of one. The old Egyptians provided sums of unit fractions as a means of expressing an ext general rational numbers — resulting in the hatchet Egyptian fractions.

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In fact, across a wide variety of mathematics we see an attention in summing unit fractions. Plenty of constants room expressed together the limitless sum that unit fractions, consisting of Apery’s constant and the reciprocal Fibonacci constant. We also see unit fractions appearing in a variety of results in statistics and in physics.

In general, any kind of unit fraction can be decomposed into the sum of two unique unit fractions as follows:


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For example 1/2 = 1/3 + 1/6, or 1/3 = 1/4 + 1/12. However, this decomposition isn’t always unique. For instance 1/4 = 1/6+1/12 = 1/5+1/20. In this short article I look at at precisely when a fraction can it is in decomposed into a distinctive sum that two distinctive unit fractions.

1. Decomposing unit fractions

Definition 1.1: A unit fraction is a rational number of the type 1/n where n > 1.

Theorem 1.2: A unit fraction can constantly be expressed as the amount of two distinctive unit fractions.

Proof:


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Theorem 1.3: For p > 1, 1/p can be expressed as a distinct sum the two unique unit fountain if and also only if p is prime.

Proof: We recognize from organize 1.2 the 1/p have the right to be expressed together follows:


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If p is prime then only one pair of unique values exist for a and b corresponding to a﹣ ns = 1 and also b ﹣ p = p² (or angry versa), and those room the worths we understand from to organize 1.2. If p is no prime climate there is much more than one pair of unique values because that a and b.

2. Decomposing fountain of the kind 2/n

In a similar method it is possible to present the same result for fractions of the type 2/p.

Theorem 2.1: For p > 2, 2/p have the right to be expressed as a distinct sum the two distinctive unit fountain if and also only if p is a prime.

Proof: Following the exact same procedure as we did in the evidence of to organize 1.3 us arrive at this identity:


Again, comparable to theorem 1.3, unique distinct values for a and b can just exist in the case where p is prime. In this case:


Note that since p is odd, these denominators will always be integers.

Example 2.2: 2/3 = 1/2 + 1/6 by this decomposition, and also no various other such unique decomposition exists.

3. Finalizing the general result

Theorem 3.1: Any (simplest form) fraction q/p much less than one deserve to be expressed together a distinct sum the two distinct unit fractions if and only if

p is prime and q = 1 orp and q are both prime and also q divides p + 1.

Proof: We have proved the case q = 1 in to organize 1.3. So i think q > 1. Again adhering to a similar method to vault proofs, we have the right to arrive at:


For unique distinctive solutions because that a and b, p and q must it is in prime. In this case, the equipment would be:


But plainly these deserve to only it is in unit fountain if q divides p + 1.

Corollary 3.2: Theorem 2.1 complies with from this more general result since if q = 2 and also p is a prime better than 2, then p is odd, so q must division p + 1.

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Example 3.3:

3/11 = 1/4 + 1/44 and there is no other means of express 3/11 together the amount of two distinctive unit fractions.97/193 =1/2 + 1/386 and there is no other way of express 97/193 together the amount of two distinctive unit fractions.
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