mathematics

Primary pseudoperfect number

Primary pseudoperfect number 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 Primary pseudoperfect number rather than just read about it. In short: In mathematics, and particularly in number theory, N is a primary pseudoperfect number if it satisfies the Egyptian fraction equation 1 N + ∑ p | N 1 p = 1 , {\displaystyle {\frac {1}{N}}+\sum _{p\,|\;\!N}{\frac {1}{p}}=1,} where the sum is over only the prime divisors of N. Properties Equivalently, N is a primary pseudoperfect number if it satisfies 1 + ∑ p | N N p = N . {\displaystyle 1+\sum _{p\,|\;\!N}{\frac {N}…

Key takeaways

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

Reference excerpt

In mathematics, and particularly in number theory, N is a primary pseudoperfect number if it satisfies the Egyptian fraction equation

1 N + ∑ p | N 1 p = 1 , {\displaystyle {\frac {1}{N}}+\sum _{p\,|\;\!N}{\frac {1}{p}}=1,}

where the sum is over only the prime divisors of N.

Properties Equivalently, N is a primary pseudoperfect number if it satisfies

1 + ∑ p | N N p = N . {\displaystyle 1+\sum _{p\,|\;\!N}{\frac {N}{p}}=N.}

Except for the primary pseudoperfect number N = 2, this expression gives a representation for N as the sum of distinct divisors of N. Therefore, each primary pseudoperfect number N (except N = 2) is also pseudoperfect. The ten known primary pseudoperfect numbers are

2, 6, 42, 1806, 47058, 2214502422, 52495396602, 5998279018951962402, 8490421583559688410706771261086, 35979351189199316534587473905773572006 (sequence A054377 in the OEIS). The first four of these numbers are one less than the corresponding numbers in Sylvester's sequence, but then the two sequences diverge. The two largest known primary pseudoperfect numbers may be out of order. It is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are any odd primary pseudoperfect numbers. The prime factors of primary pseudoperfect numbers sometimes may provide solutions to Znám's problem, in which all elements of the solution set are prime. For instance, the prime factors of the primary pseudoperfect number 47058 form the solution set {2,3,11,23,31} to Znám's problem. However, the smaller primary pseudoperfect numbers 2, 6, 42, and 1806 do not correspond to solutions to Znám's problem in this way, as their sets of prime factors violate the requirement that no number in the set can equal one plus the product of the other numbers. Anne (1998) observes that there is exactly one solution set of this type that has k primes in it, for each k ≤ 8, and conjectures that the same is true for larger k. If a primary pseudoperfect number N is one less than a prime number, then N × (N + 1) is also primary pseudoperfect. For instance, 47058 is primary pseudoperfect, and 47059 is prime, so 47058 × 47059 = 2214502422 is also primary pseudoperfect.

History Primary pseudoperfect numbers were first investigated and named by Butske, Jaje & Mayernik (2000). Using computational search techniques, they proved the remarkable result that for each positive integer r up to 8, there exists exactly one primary pseudoperfect number with precisely r (distinct) prime factors, namely, the rth known primary pseudoperfect number. Those with 2 ≤ r ≤ 8, when reduced modulo 288, form the arithmetic progression 6, 42, 78, 114, 150, 186, 222, as was observed by Sondow & MacMillan (2017). The ninth known primary pseudoperfect number was discovered by Wang (2026) and Martins, Pedro (2026).

Illustrations

Worked examples

Example 1 — a first encounter with Primary pseudoperfect number

Start with the simplest possible case. Write down what Primary pseudoperfect number 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 Primary pseudoperfect number 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 Primary pseudoperfect number 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 Primary pseudoperfect number

In research: Primary pseudoperfect number 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 Primary pseudoperfect number 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: Primary pseudoperfect number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Egyptian fractions, Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Primary pseudoperfect number 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 Primary pseudoperfect number in 20 minutes

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

Frequently asked questions

What is Primary pseudoperfect number in simple terms?

In mathematics, and particularly in number theory, N is a primary pseudoperfect number if it satisfies the Egyptian fraction equation 1 N + ∑ p | N 1 p = 1 , {\displaystyle {\frac {1}{N}}+\sum _{p\,|\;\!N}{\frac {1}{p}}=1,} where the sum is over only the prime divisors of N. Properties Equivalently…

Why does Primary pseudoperfect number 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 Primary pseudoperfect number?

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 Primary pseudoperfect number.