Gilbreath's conjecture

Key takeaways

Gilbreath's conjecture belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Gilbreath's conjecture to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Gilbreath's conjecture from memory before moving on to harder problems.

Reference excerpt

Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The statement is named after Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the pattern by chance while doing arithmetic on a napkin. In 1878, eighty years before Gilbreath's discovery, François Proth had published the same observations.

Motivating arithmetic Consider the prime numbers

2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , … {\displaystyle 2,3,5,7,11,13,17,19,23,29,31,\dots }

Computing the absolute value of the difference between term n {\displaystyle n} and term n + 1 {\displaystyle n+1} in this sequence yields the sequence

1 , 2 , 2 , 4 , 2 , 4 , 2 , 4 , 6 , 2 , … {\displaystyle 1,2,2,4,2,4,2,4,6,2,\dots }

If the same calculation is done for the terms in this new sequence, and the sequence that is the outcome of this process, and again ad infinitum for each sequence that is the output of such a calculation, the following five sequences in this list are

1 , 0 , 2 , 2 , 2 , 2 , 2 , 2 , 4 , … {\displaystyle 1,0,2,2,2,2,2,2,4,\dots }

1 , 2 , 0 , 0 , 0 , 0 , 0 , 2 , … {\displaystyle 1,2,0,0,0,0,0,2,\dots }

1 , 2 , 0 , 0 , 0 , 0 , 2 , … {\displaystyle 1,2,0,0,0,0,2,\dots }

1 , 2 , 0 , 0 , 0 , 2 , … {\displaystyle 1,2,0,0,0,2,\dots }

1 , 2 , 0 , 0 , 2 , … {\displaystyle 1,2,0,0,2,\dots }

What Gilbreath—and François Proth before him—noticed is that the first term in each series of differences appears to be 1.

The conjecture Formally, let ( p n ) {\displaystyle (p_{n})} denote the sequence of prime numbers. We can define the sequence ( d n k ) {\displaystyle (d_{n}^{k})} recursively by

d n k = { p n + 1 − p n , k = 1 | d n + 1 k − 1 − d n k − 1 | , k ≥ 1. {\displaystyle d_{n}^{k}={\begin{cases}p_{n+1}-p_{n},&k=1\\[5pt]|d_{n+1}^{k-1}-d_{n}^{k-1}|,&k\geq 1.\end{cases}}}

Gilbreath's conjecture states that d 1 k = 1 {\displaystyle d_{1}^{k}=1} for all k ≥ 1 {\displaystyle k\geq 1} .

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Gilbreath's conjecture

Start with the simplest possible case. Write down what Gilbreath's conjecture claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Gilbreath's conjecture before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Gilbreath's conjecture ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Gilbreath's conjecture

In research: Gilbreath's conjecture appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Gilbreath's conjecture in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Gilbreath's conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Analytic number theory, Conjectures about prime numbers, Triangles of numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Gilbreath's conjecture outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Gilbreath's conjecture in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Gilbreath's conjecture means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Gilbreath's conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Gilbreath's conjecture in simple terms?

Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The stat…

Why does Gilbreath's conjecture matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Gilbreath's conjecture?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Gilbreath's conjecture.