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Pentagonal number

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Figurate number

Not to be confused withcentered pentagonal number.

A visual representation of the first six pentagonal numbers

Apentagonal number is afigurate number that extends the concept oftriangular andsquare numbers to thepentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are notrotationally symmetrical. Thenth pentagonal numberp_n is the number ofdistinct dots in a pattern of dots consisting of theoutlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share onevertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside.

p_n is given by the formula:

p_{n}={\frac {3n^{2}-n}{2}}={\binom {n}{1}}+3{\binom {n}{2}}

forn ≥ 1. The first few pentagonal numbers are:

1,5,12,22,35,51,70,92,117,145,176,210,247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925,1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187... (sequenceA000326 in theOEIS).

Thenth pentagonal number is the sum of n integers starting from n (i.e. from n to 2n − 1). The following relationships also hold:

p_{n}=p_{n-1}+3n-2=2p_{n-1}-p_{n-2}+3

Pentagonal numbers are closely related to triangular numbers. Thenth pentagonal number is one third of the(3n − 1)thtriangular number. In addition, where T_n is thenth triangular number:

p_{n}=T_{n-1}+n^{2}=T_{n}+2T_{n-1}=T_{2n-1}-T_{n-1}

Generalized pentagonal numbers are obtained from the formula given above, but withn taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4..., producing the sequence:

0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335... (sequenceA001318 in theOEIS).

Generalized pentagonal numbers are important toEuler's theory ofinteger partitions, as expressed in hispentagonal number theorem.

The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.

Other properties

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$p_{n}$ for n>0 is the number of differentcompositions of $n+8$ into n parts that don't include 2 or 3.
$p_{n}$ is the sum of the first n natural numbers congruent to 1 mod 3.
$p_{8n}-p_{8n-1}=p_{2n+2}-p_{2n-2}$

Generalized pentagonal numbers and centered hexagonal numbers

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Generalized pentagonal numbers are closely related tocentered hexagonal numbers. When the array corresponding to a centered hexagonal number is divided between its middle row and an adjacent row, it appears as the sum of two generalized pentagonal numbers, with the larger piece being a pentagonal number proper:

1=1+0		7=5+2		19=12+7		37=22+15

In general:

3n(n-1)+1={\tfrac {1}{2}}n(3n-1)+{\tfrac {1}{2}}(1-n){\bigl (}3(1-n)-1{\bigr )}

where both terms on the right are generalized pentagonal numbers and the first term is a pentagonal number proper (n ≥ 1). This division of centered hexagonal arrays gives generalized pentagonal numbers as trapezoidal arrays, which may be interpreted as Ferrers diagrams for their partition. In this way they can be used to prove the pentagonal number theorem referenced above.

Proof without words that thenth pentagonal number can be decomposed into three equal (n-1)thtriangular numbers and the numbern.

Sum of reciprocals

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A formula for thesum of the reciprocals of the pentagonal numbers is given by $\sum _{n=1}^{\infty }{\frac {2}{n\left(3n-1\right)}}=3\ln \left(3\right)-{\frac {\pi }{\sqrt {3}}}.$

Tests for pentagonal numbers

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Given a positive integerx, to test whether it is a (non-generalized) pentagonal number we can compute

n={\frac {{\sqrt {24x+1}}+1}{6}}.

The numberx is pentagonal if and only ifn is anatural number. In that casex is thenth pentagonal number.

For generalized pentagonal numbers, it is sufficient to just check if24x + 1 is a perfect square.

For non-generalized pentagonal numbers, in addition to the perfect square test, it is also required to check if

{\sqrt {24x+1}}\equiv 5\mod 6

The mathematical properties of pentagonal numbers ensure that these tests are sufficient for proving or disproving the pentagonality of a number.^[1]

Gnomon

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TheGnomon of thenth pentagonal number is:

p_{n+1}-p_{n}=3n+1

Square pentagonal numbers

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A square pentagonal number is a pentagonal number that is also a perfect square.^[2]

The first few are:

0, 1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, 7681419682192581869134354401, 73756990988431941623299373152801... (OEIS entryA036353)

Pentagonal Square Triangular Number

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In number theory, a pentagonal square triangular number is a positive integer that is simultaneously a pentagonal number, a square number, and a triangular number. This requires solving the following system ofDiophantine equations

P_{n}={\frac {n(3n-1)}{2}}=T_{m}={\frac {m(m+1)}{2}}=k^{2}

where $P_{n}$ is the $n {\displaystyle n}$ -th pentagonal number, $T_{m}$ is the $m {\displaystyle m}$ -th triangular number, and $k^{2}$ is a square number.

Solutions to this problem can be found by checking pentagonal triangular numbers against square numbers. Other than the trivial solution of 1, computational searches of the first 9,690 pentagonal triangular numbers have revealed no other square numbers, suggesting that no other pentagonal square triangular numbers exist below this limit.^[3]

Although no formal proof has yet appeared in print, work by J. Sillcox between 2003 and 2006 applied results from W. S. Anglin's 1996 paper on simultaneous Pell equations to this problem. Anglin demonstrated that simultaneous Pell equations have exactly 19,900 solutions with $x,y<10^{20000}$ .^[4] Sillcox showed that the pentagonal square triangular number problem can be reduced to solving the equation:

x^{2}-6y^{2}=-5

This places the problem within the scope of Anglin's proof. For $x=1$ and $y=1$ , only the trivial solution exists.^[3]^[4]

References

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^How do you determine if a number N is a Pentagonal Number?
^Weisstein, Eric W. "Pentagonal Square Number." FromMathWorld--A Wolfram Web Resource.
^^a ^bWeisstein, Eric W."Pentagonal Square Triangular Number".MathWorld. Wolfram. RetrievedApril 2, 2025.
^^a ^bAnglin, W.S. (1996). "Simultaneous Pell Equations".Math. Comput.65 (213):355–359.doi:10.1090/S0025-5718-96-00687-4.

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