Pentagonal Number Checker

P(n) = n(3n−1)/2

CalculatorsFreeNo Signup
4.5(606 reviews)
All Tools

Loading tool...

About Pentagonal Number Checker

A pentagonal number checker testing if n = k(3k−1)/2. Shows generalized pentagonal numbers (k can be negative), connects to Euler's pentagonal theorem for partition function. Scans nearby pentagons. All client-side.

Pentagonal Number Checker Features

  • Pentagon check
  • Generalized
  • Euler theorem
  • Sequence
  • Range scan
Pentagonal numbers: P(n) = n(3n−1)/2. Sequence: 1,5,12,22,35,51,70... Generalized: allow negative k, giving 1,2,5,7,12,15,22,26,35... Euler's pentagonal theorem: Πₙ(1−xⁿ) = Σₖ(−1)ᵏx^(k(3k−1)/2).

How to Use

Enter n:

  • Check: Is n pentagonal?
  • Index: Solve for k
  • Generalized: Extended sequence

Euler's Pentagonal Theorem

Π(1−xⁿ) = 1−x−x²+x⁵+x⁷−x¹²−x¹⁵+... Exponents are generalized pentagonal numbers. Signs: +,−,−,+,+,−,−,+,+... This gives a recurrence for the partition function p(n).

Properties

  • Sum formula: ΣP(k) = n²(n+1)/2
  • Test: n pentagonal iff (1+√(1+24n))/6 is integer
  • Three pentagonal numbers suffice (Gauss)

Step-by-Step Instructions

  1. 1Enter n.
  2. 2Check pentagonal.
  3. 3Find k.
  4. 4View generalized.
  5. 5See Euler's theorem.

Pentagonal Number Checker — Frequently Asked Questions

What are generalized pentagonal numbers?+

Allow k to be 0, ±1, ±2, ±3,...: P(k) = k(3k−1)/2 gives 0,1,2,5,7,12,15,22,26,35,40... These are the exponents in Euler's pentagonal theorem and crucial for computing the partition function recursively.

How does Euler's theorem help compute partitions?+

p(n) = p(n−1) + p(n−2) − p(n−5) − p(n−7) + p(n−12) + ... The indices subtracted are generalized pentagonal numbers, and signs alternate in pairs (−,−,+,+,−,−,...). This is the fastest known elementary recurrence for partitions.

Can every number be written as sum of pentagonal numbers?+

Yes! By Fermat's polygonal number theorem (proved by Cauchy), every positive integer is the sum of at most 5 pentagonal numbers. In practice, most need fewer. Every sufficiently large n needs at most 3 pentagonal numbers.

More Calculators Tools

Explore all 639 other calculators tools available on Generatr

1-Sample Z-Test Calculator2-Sample Z-Test Calculator3D Surface Area CalculatorA/B Testing Significance CalculatorAbsolute Value CalculatorAbundant Number CheckerAccelerated Aging CalculatorAchromatic Number CalculatorAcyclic Chromatic Number CalculatorAdditive Persistence CalculatorAge CalculatorAging Test CalculatorAlbertson Index CalculatorAlgebraic Connectivity CalculatorAliquot Sequence CalculatorAliquot Sum CalculatorAlternating Series Test CalculatorAmicable Number FinderAnnuity CalculatorAPY CalculatorAquarium CO2 Drop CheckerArea CalculatorArea Under Curve CalculatorArithmetic Sequence CalculatorArmstrong Number CalculatorASCVD Risk CalculatorAspect Ratio CalculatorAtom Bond Connectivity CalculatorAugmented Eccentric Connectivity CalculatorAugmented Zagreb Index Calculator
View all 639 calculators tools
Share this tool: