A palindromic prime is a prime number whose digits read the same from left to right and right to left. In base 10, examples include 2, 11, 101, 131, and 929. A number must satisfy both conditions: its digit sequence must be a palindrome, and it must have exactly two positive divisors.
Palindromic Prime Definition
A palindromic prime is a whole number greater than 1 that is prime and remains unchanged when its digits are reversed.
What Makes a Number a Palindromic Prime?
Palindromic primes combine two separate mathematical properties. The palindrome condition concerns the number’s written digits, while primality concerns its divisors.
Palindrome
The digits appear in the same order when read in either direction. For example, 131 reverses to 131.
Prime Number
The number has exactly two positive divisors: 1 and the number itself.
Both Properties
131 is palindromic and has no positive divisors other than 1 and 131.
Palindrome but Composite
121 reads both ways, but 121 = 11 × 11, so it is composite.
A palindrome is therefore only a prime candidate. Its symmetrical digits do not prove that it is prime. A separate primality test is still needed.
Palindromic Primes Below 1,000
There are 20 palindromic primes in the inclusive range from 2 through 999. The one-digit values qualify automatically as palindromes because reversing a single digit does not change it.
| Number of Digits | Palindromic Primes | Count |
|---|---|---|
| 1 digit | 2, 3, 5, 7 | 4 |
| 2 digits | 11 | 1 |
| 3 digits | 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929 | 15 |
The sequence continues with five-digit values such as 10301, 10501, 10601, 11311, and 11411. There are no four-digit palindromic primes because every four-digit palindrome is divisible by 11.
Why 11 Is the Even-Digit Exception
Every base-10 palindrome containing an even number of digits is divisible by 11. This follows from the divisibility rule for 11: the alternating sum of a number’s digits must be a multiple of 11.
Example with 1221
1 − 2 + 2 − 1 = 0
Because 0 is divisible by 11, 1221 is also divisible by 11. In fact, 1221 = 11 × 111.
In an even-length palindrome, mirrored digits occupy alternating positions and cancel one another. The alternating sum is therefore 0, making the number divisible by 11.
Digit-Length Rule
11 is the only palindromic prime with an even number of decimal digits. Every larger even-digit palindrome has 11 as a proper factor and is composite. All base-10 palindromic primes greater than 11 consequently have an odd number of digits.
Palindrome and Prime Are Different Properties
Reversing digits and testing divisibility answer different questions. A number may have either property, both properties, or neither.
| Number | Palindrome? | Prime? | Reason |
|---|---|---|---|
| 131 | Yes | Yes | Its only positive divisors are 1 and 131. |
| 121 | Yes | No | 121 = 11 × 11. |
| 97 | No | Yes | 97 is prime, but its reversal is 79. |
| 120 | No | No | It is not symmetrical and has several positive divisors. |
Palindromic Does Not Mean Prime
Numbers such as 22, 121, 1001, and 12321 are palindromes, but each is composite. A digit pattern can identify a candidate; it cannot replace a primality test.
Checking a Palindromic Candidate
A candidate first passes the digit test: its usual decimal representation must match its reversal. Primality can then be tested through divisibility.
For trial division, possible prime divisors only need to be checked up to √n. This works because every composite number has a factor pair. If both factors were greater than √n, their product would be greater than n.
Testing 101
√101 is slightly greater than 10. The relevant prime divisors are 2, 3, 5, and 7. None divides 101, so 101 is prime.
Testing 121
√121 = 11, and 121 is divisible by 11. Its factor pair 11 × 11 proves that it is composite.
Test a Palindromic Number
A palindrome test examines the arrangement of digits. The prime number checker tests whether the value is prime, composite, or outside those two classifications.
Test a Palindromic Candidate
Enter a whole number and check whether it is prime, composite, or neither.
The Numeral Base Matters
Palindromicity depends on how a number is written. The term usually refers to the standard base-10 representation unless another numeral system is named.
For example, 17 is written as 17 in decimal, so it is not a decimal palindrome. In binary, the same value is written as 10001, which is palindromic. Its primality does not change—the representation changes, not the number’s positive divisors.
Representation and Divisibility
Prime is an intrinsic property of a positive integer. Palindromic is a property of its digit representation in a chosen base.
Related Number Theory Concepts
Frequently Asked Questions
Is Every Palindrome a Prime Number?
No. A palindrome only has matching forward and backward digit sequences. For example, 121 is palindromic but composite because 121 = 11 × 11.
Are All Single-Digit Primes Palindromic?
Yes. The single-digit primes are 2, 3, 5, and 7. Each consists of one digit, so its reversal is unchanged.
Is 11 a Palindromic Prime?
Yes. The digits of 11 read the same in either direction, and its only positive divisors are 1 and 11.
Why Are There No Four-Digit Palindromic Primes?
Every four-digit palindrome is divisible by 11. Since every four-digit palindrome is greater than 11, that divisor makes it composite.
Is 101 a Palindromic Prime?
Yes. Reversing 101 gives 101, and none of the primes up to √101—2, 3, 5, or 7—divides it.
Does a Palindromic Prime Have to Be Odd?
Every palindromic prime except 2 is odd. This follows from the broader rule that 2 is the only even prime number.
Is a Palindromic Number Automatically a Prime Candidate?
It can be considered a candidate, but many palindromes can be rejected immediately through divisibility. Every even-digit decimal palindrome greater than 11, for example, is divisible by 11.