Primefree sequence

Key takeaways

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

Reference excerpt

In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fibonacci numbers, but with different initial conditions causing all members of the sequence to be composite numbers that do not all have a common divisor. To put it algebraically, a sequence of this type is defined by an appropriate choice of two composite numbers a1 and a2, such that the greatest common divisor g c d ( a 1 , a 2 ) {\displaystyle \mathrm {gcd} (a_{1},a_{2})} is equal to 1, and such that for n > 2 {\displaystyle n>2} there are no primes in the sequence of numbers calculated from the formula

a n = a n − 1 + a n − 2 {\displaystyle a_{n}=a_{n-1}+a_{n-2}} . The first primefree sequence of this type was published by Ronald Graham in 1964.

Wilf's sequence A primefree sequence found by Herbert Wilf has initial terms

a 1 = 20615674205555510 , a 2 = 3794765361567513 {\displaystyle a_{1}=20615674205555510,a_{2}=3794765361567513} (sequence A083216 in the OEIS) The proof that every term of this sequence is composite relies on the periodicity of Fibonacci-like number sequences modulo the members of a finite set of primes. For each prime p {\displaystyle p} , the positions in the sequence where the numbers are divisible by p {\displaystyle p} repeat in a periodic pattern, and different primes in the set have overlapping patterns that result in a covering set for the whole sequence.

Nontriviality The requirement that the initial terms of a primefree sequence be coprime is necessary for the question to be non-trivial. If the initial terms share a prime factor p {\displaystyle p} (e.g., set a 1 = x p {\displaystyle a_{1}=xp} and a 2 = y p {\displaystyle a_{2}=yp} for some x {\displaystyle x} and y {\displaystyle y} both greater than 1), due to the distributive property of multiplication a 3 = ( x + y ) p {\displaystyle a_{3}=(x+y)p} and more generally all subsequent values in the sequence will be multiples of p {\displaystyle p} . In this case, all the numbers in the sequence will be composite, but for a trivial reason. The order of the initial terms is also important. In Paul Hoffman's biography of Paul Erdős, The man who loved only numbers, the Wilf sequence is cited but with the initial terms switched. The resulting sequence appears primefree for the first hundred terms or so, but term 138 is the 45-digit prime 439351292910452432574786963588089477522344721 {\displaystyle 439351292910452432574786963588089477522344721} .

Other sequences Several other primefree sequences are known:

a 1 = 331635635998274737472200656430763 , a 2 = 1510028911088401971189590305498785 {\displaystyle a_{1}=331635635998274737472200656430763,a_{2}=1510028911088401971189590305498785} (sequence A083104 in the OEIS; Graham 1964),

a 1 = 62638280004239857 , a 2 = 49463435743205655 {\displaystyle a_{1}=62638280004239857,a_{2}=49463435743205655} (sequence A083105 in the OEIS; Knuth 1990), and

a 1 = 407389224418 , a 2 = 76343678551 {\displaystyle a_{1}=407389224418,a_{2}=76343678551} (sequence A082411 in the OEIS; Nicol 1999). The sequence of this type with the smallest known initial terms has

a 1 = 106276436867 , a 2 = 35256392432 {\displaystyle a_{1}=106276436867,a_{2}=35256392432} (sequence A221286 in the OEIS; Vsemirnov 2004).

Notes

References Graham, Ronald L. (1964). "A Fibonacci-like sequence of composite numbers" (PDF). Mathematics Magazine. 37 (5): 322–324. doi:10.2307/2689243. JSTOR 2689243. Knuth, Donald E. (1990). "A Fibonacci-like sequence of composite numbers". Mathematics Magazine. 63 (1): 21–25. doi:10.2307/2691504. JSTOR 2691504. MR 1042933. Wilf, Herbert S. (1990). "Letters to the Editor". Mathematics Magazine. 63: 284. doi:10.1080/0025570X.1990.11977539. JSTOR 2690956. Nicol, John W. (1999). "A Fibonacci-like sequence of composite numbers" (PDF). Electronic Journal of Combinatorics. 6 (1): 44. doi:10.37236/1476. MR 1728014. Vsemirnov, M. (2004). "A new Fibonacci-like sequence of composite numbers" (PDF). Journal of Integer Sequences. 7 (3): 04.3.7. Bibcode:2004JIntS...7...37V. MR 2110778.

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Worked examples

Example 1 — a first encounter with Primefree sequence

Start with the simplest possible case. Write down what Primefree sequence 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 Primefree sequence 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 Primefree sequence 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 Primefree sequence

In research: Primefree sequence 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 Primefree sequence 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: Primefree sequence is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, Number theory, Recurrence relations, so understanding it makes those chapters shorter.
In everyday life: Look for Primefree sequence 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 Primefree sequence in 20 minutes

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

Frequently asked questions

What is Primefree sequence in simple terms?

In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fibonacci numbers, but with different initial conditions causing all members of the sequence to be co…

Why does Primefree sequence 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 Primefree sequence?

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 Primefree sequence.