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Armaan Nougai

The nth term of the *pentagonal sequence* is defined as:

```
Pn = n*(3*n - 1)/2
```

Find a pair of two pentagonal numbers `Pj`

and `Pk`

, such that:

```
Pk + Pj = S is a pentagonal
number
Pk - Pj = D is a pentagonal
number and is minimum too.
```

What is the value of `D`

?

Question analysis

We will run two loops:

- an outer loop over pentagonal sequence for the value of
`P(k)`

. - an inner loop over pentagonal sequence less than
`P(k)`

for`P(j)`

.

Then, we will calculate and check if `D`

and `S`

are pentagonal.
To check this, we will inverse the pentagonal function.

The moment we get a pair for which `D`

and `S`

are pentagonal, we’ll break the loop and return `D`

.
This is because the first pentagonal `D`

is also the minimum one.

Inverse of pentagonal function

""" Coded by - Armaan Nougai """ from math import sqrt def is_pentagonal(n): return (1+sqrt(1+24*n))%6==0 i=0 while True: i+=1 k = i*(3*i-1)//2 for v in range(1,i): j = v*(3*v-1)//2 if is_pentagonal(k-j) and is_pentagonal(k+j) : print(k-j) break else: continue break

Why is the first value of`D`

(i.e., 5482660) the minimum possible value of`D`

?

On analyzing the pentagonal sequence:

**1.** `P(2)-P(1)`

= `4`

**2.** `P(3)-P(2)`

= `7`

**3.** `P(4)-P(3)`

= `10`

```
P(n)-P(n-1) < P(n+1)-P(n) < P(n+2)-P(n+1) ...
```

This means the difference between adjacent terms is increasing.

And also, for the nth term, the minimum `D`

, even without the condition, will be:

```
P(n)-P(n-1)
```

Now, after getting the first value of `D`

, we will continue the search for the lesser value of `D`

until we find an *adjacent pair* that satisfies the condition and whose `D`

is *greater than* the first value of `D`

.

This is because, after this point, the `D`

of every pair will be greater than the first value of `D`

.

RELATED TAGS

pentagonal numbers

project euler

project euler 44

solution

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Armaan Nougai

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