The **Euler's Totient Function** (*phi function*) Ο of a palindromic prime *p* is easy to compute.

**Definition of Terms**:

**Euler's Phi Function**- a number of positive integers less than or equal to *n* that are relatively prime to *n*.

**Palindrome**- a word, a phrase, a sequence, or a number that reads the same backward as forward.

**Prime**- a natural number greater than 1 that has no positive divisors other than 1 and itself.

**Palindromic prime**- a number that is simultaneously palindromic and prime.

** Conjecture**:

The phi function Ο of every palindromic prime *p* is equal to a given palindromic prime *p* minus 1.

In this conjecture, let *n* equals *p*.

**Note**: *Palindromic prime, prime palindrome and palprime are synonymous.* π

In symbols:

Ο(*p*)= *p*-1 (**FORMULA**) ππ¦

**Example 1**:

*p*= **929** (*a palindromic prime*)

Solution:

Use the formula.

```
Ο(929) = 929-1
= 928 βπ
```

Hence, Ο(**929**) = **928**.

**Example 2**:

*p*= **13,331** (*a palindromic prime*)

Solution:

Use the formula.

```
Ο(13331) = 13331-1
= 13,330 βπ
```

Hence, Ο(**13331**) = **13,330**.

Now, you try! π

**Exercises** π

Compute the following:

```
1. Ο(10301)
2. Ο(1411141)
3. Ο(7619167)
4. Ο(7630367)
5. Ο(9989899)
```

Author: **John Paul L. Hablado**, *LPT*

```
(c) April 11, 2017
```

References:

Kindly click each **example link** for the **URL** (Uniform Resource Locator), and some related theorems on **Euler's Phi Function of a Palindromic Prime**. β€

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