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In a recent paper in Integers, Hannah Alpert proves that "every integer can be written uniquely as a sum of Fibonacci numbers and their additive inverses, such that every two terms of the same sign differ in index by at least 4 and every two terms of different sign differ in index by at least 3. Furthermore, there is no way to use fewer terms to write a number as a sum of Fibonacci numbers and their additive inverses." Alpert was at the Duluth REU mentioned by Kevin O'Bryant in his commenthis comment; I suspect this is the paper referred to and the result was at some point miscommunicated.

The procedure Alpert gives for finding such a representaiton gives, for example, 5473 = 6765 - 1597 + 377 - 89 + 21 - 5 + 1. I think 5473 is the smallest number which needs seven terms.

In a recent paper in Integers, Hannah Alpert proves that "every integer can be written uniquely as a sum of Fibonacci numbers and their additive inverses, such that every two terms of the same sign differ in index by at least 4 and every two terms of different sign differ in index by at least 3. Furthermore, there is no way to use fewer terms to write a number as a sum of Fibonacci numbers and their additive inverses." Alpert was at the Duluth REU mentioned by Kevin O'Bryant in his comment; I suspect this is the paper referred to and the result was at some point miscommunicated.

The procedure Alpert gives for finding such a representaiton gives, for example, 5473 = 6765 - 1597 + 377 - 89 + 21 - 5 + 1. I think 5473 is the smallest number which needs seven terms.

In a recent paper in Integers, Hannah Alpert proves that "every integer can be written uniquely as a sum of Fibonacci numbers and their additive inverses, such that every two terms of the same sign differ in index by at least 4 and every two terms of different sign differ in index by at least 3. Furthermore, there is no way to use fewer terms to write a number as a sum of Fibonacci numbers and their additive inverses." Alpert was at the Duluth REU mentioned by Kevin O'Bryant in his comment; I suspect this is the paper referred to and the result was at some point miscommunicated.

The procedure Alpert gives for finding such a representaiton gives, for example, 5473 = 6765 - 1597 + 377 - 89 + 21 - 5 + 1. I think 5473 is the smallest number which needs seven terms.

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Michael Lugo
Michael Lugo
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In a recent paper in Integers, Hannah Alpert proves that "every integer can be written uniquely as a sum of Fibonacci numbers and their additive inverses, such that every two terms of the same sign differ in index by at least 4 and every two terms of different sign differ in index by at least 3. Furthermore, there is no way to use fewer terms to write a number as a sum of Fibonacci numbers and their additive inverses." Alpert was at the Duluth REU mentioned by Kevin O'Bryant in his comment; I suspect this is the paper referred to and the result was at some point miscommunicated.

The procedure Alpert gives for finding such a representaiton gives, for example, 5473 = 6765 - 1597 + 377 - 89 + 21 - 5 + 1. I think 5473 is the smallest number which needs seven terms.