A unit fraction is a portion with a molecule of one. The old Egyptians used sums that unit fractions together a way of expressing more general rational number — causing the ax Egyptian fractions.

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In fact, across a wide range of mathematics we watch an attention in summing unit fractions. Numerous constants space expressed as the limitless sum the unit fractions, including Apery’s continuous and the mutual Fibonacci constant. We likewise see unit fractions showing up in a number of results in statistics and also in physics.

In general, any kind of unit portion can it is in decomposed into the amount of two distinct 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 constantly unique. For example 1/4 = 1/6+1/12 = 1/5+1/20. In this write-up I look at specifically when a portion can it is in decomposed into a unique sum of two distinctive unit fractions.

1. Decomposing unit fractions

Definition 1.1: A unit portion is a rational variety of the kind 1/n whereby n > 1.

Theorem 1.2: A unit fraction can constantly be expressed together the sum of two distinct unit fractions.

Proof:


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Theorem 1.3: For p > 1, 1/p have the right to be expressed together a distinct sum of two unique unit fractions if and also only if p is prime.

Proof: We understand from theorem 1.2 the 1/p can be expressed as follows:


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If p is element then only one pair of distinct values exist because that a and b corresponding come a﹣ ns = 1 and b ﹣ ns = p² (or evil versa), and those are the values we recognize from organize 1.2. If p is no prime climate there is much more than one pair of distinctive values for a and b.

2. Decomposing fractions of the form 2/n

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

Theorem 2.1: For p > 2, 2/p deserve to be expressed as a distinct sum that two unique unit fractions if and also only if p is a prime.

Proof: Following the exact same procedure together we did in the proof of organize 1.3 us arrive in ~ this identity:


Again, similar to to organize 1.3, unique unique values for a and b can only exist in the case where p is prime. In this case:


Note that due to the fact that p is odd, these denominators will always be integers.

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

3. Finalizing the general result

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

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

Proof: We have actually proved the instance q = 1 in theorem 1.3. So assume q > 1. Again following a similar an approach to vault proofs, we can arrive at:


For unique distinct solutions because that a and b, ns and q must be prime. In this case, the systems would be:


But plainly these can only it is in unit fractions if q divides p + 1.

Corollary 3.2: Theorem 2.1 complies with from this an ext general result since if q = 2 and p is a prime better than 2, climate p is odd, so q must divide p + 1.

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Example 3.3:

3/11 = 1/4 + 1/44 and there is no other method of to express 3/11 together the amount of two distinctive unit fractions.97/193 =1/2 + 1/386 and there is no other method of expressing 97/193 as the amount of two distinct unit fractions.
Expert and also Author in used Mathematics, Data Science, Statistics and Psychometrics. Find me top top LinkedIn or Twitter or drkeithmcnulty.com


Expert and Author in used Mathematics, Data Science, Statistics and Psychometrics. Uncover me top top LinkedIn or Twitter or drkeithmcnulty.com