mathematics

Scholz conjecture

Scholz conjecture is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Scholz conjecture rather than just read about it. In short: In mathematics, the Scholz conjecture is a conjecture on the length of certain addition chains. It is sometimes also called the Scholz–Brauer conjecture or the Brauer–Scholz conjecture, after Arnold Scholz, who formulated it in 1937, and Alfred Brauer, who studied it soon afterward and proved a weaker bound.

Key takeaways

Scholz 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 Scholz conjecture to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Scholz conjecture from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Scholz conjecture is a conjecture on the length of certain addition chains. It is sometimes also called the Scholz–Brauer conjecture or the Brauer–Scholz conjecture, after Arnold Scholz, who formulated it in 1937, and Alfred Brauer, who studied it soon afterward and proved a weaker bound. Neill Clift has announced an example showing that the bound of the conjecture is not always tight.

Statement The conjecture states that

l(2n − 1) ≤ n − 1 + l(n), where l(n) is the length of the shortest addition chain producing n. Here, an addition chain is defined as a sequence of numbers, starting with 1, such that every number after the first can be expressed as a sum of two earlier numbers (which are allowed to both be equal). Its length is the number of sums needed to express all its numbers, which is one less than the length of the sequence of numbers (since there is no sum of previous numbers for the first number in the sequence, 1). Computing the length of the shortest addition chain that contains a given number x can be done by dynamic programming for small numbers, but it is not known whether it can be done in polynomial time measured as a function of the length of the binary representation of x. Scholz's conjecture, if true, would provide short addition chains for numbers x of a special form, the Mersenne numbers.

Example As an example, l(5) = 3: it has a shortest addition chain

1, 2, 4, 5 of length three, determined by the three sums

1 + 1 = 2, 2 + 2 = 4, 4 + 1 = 5. Also, l(31) = 7: it has a shortest addition chain

1, 2, 3, 6, 12, 24, 30, 31 of length seven, determined by the seven sums

1 + 1 = 2, 2 + 1 = 3, 3 + 3 = 6, 6 + 6 = 12, 12 + 12 = 24, 24 + 6 = 30, 30 + 1 = 31. Both l(31) and 5 − 1 + l(5) equal 7. Therefore, these values obey the inequality (which in this case is an equality) and the Scholz conjecture is true for the case n = 5.

Partial results By using a combination of computer search techniques and mathematical characterizations of optimal addition chains, Clift (2011) showed that the conjecture is true for all n < 5784689. Additionally, he verified that for all n ≤ 64, the inequality of the conjecture is actually an equality. The bound of the conjecture is not always an exact equality. For instance, for n = 9307543 {\displaystyle n=9307543} , l ( 2 n − 1 ) ≤ 9307570 < 9307571 = n − 1 + l ( n ) {\displaystyle l(2^{n}-1)\leq 9307570<9307571=n-1+l(n)} , with l ( n ) = 29 {\displaystyle l(n)=29} .

References

External links Shortest addition chains OEIS sequence A003313

Worked examples

Example 1 — a first encounter with Scholz conjecture

Start with the simplest possible case. Write down what Scholz 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 Scholz 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 Scholz 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 Scholz conjecture

In research: Scholz 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 Scholz 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: Scholz conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Addition chains, Conjectures, Unsolved problems in number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Scholz 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.

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How to study Scholz conjecture in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Scholz 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 Scholz conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Scholz conjecture in simple terms?

Why does Scholz 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 Scholz 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 Scholz conjecture.