Digitally Delicate Primes Explained: When One Digit Changes Everything

294001 is prime. Change any one of its six decimal digits to any other digit, however, and the new number is composite. A change in the hundred-thousands place, tens place, or units place makes no difference. Every permitted one-digit mutation destroys primality.

This property makes 294001 a digitally delicate prime, also called a weakly prime number in some integer-sequence references. The word “delicate” refers to the decimal representation rather than the numerical size of the prime: one changed digit is enough to make primality disappear.

What Is a Digitally Delicate Prime?

A prime number is digitally delicate in base 10 when every number formed by replacing exactly one of its written digits with another decimal digit is composite.

The definition contains several conditions:

• The original number must be prime.
• Exactly one digit is replaced.
• The replacement digit must differ from the original digit.
• Every digit position must be tested.
• All nine alternative digits must be tested at each position.
• Every resulting integer must be composite.

Digit deletion, insertion, reversal, and swapping two positions are different operations. They do not belong to the standard definition.

Composite is more precise than “not prime”

The mutated result must be composite, not merely nonprime. This distinction matters because 0 and 1 are neither prime nor composite. If an allowed digit replacement produced 0 or 1, the original prime would fail the definition.

For ordinary multi-digit candidates, this edge case rarely appears. It still belongs in a mathematically exact test.

Does changing the first digit to zero count?

Yes. The first written digit is one of the number’s positions, so changing it to 0 must be tested. The new integer is read without the leading zero.

For example:

The result 94001 is composite because:

The standard definition does not test extra zeros imagined farther to the left. That stronger condition belongs to widely digitally delicate primes.

Why 294001 Requires 54 Tests

Suppose a decimal number has n digits. Each position can be changed to any of the other nine decimal digits. The number of distinct one-digit replacements is therefore:

For 294001:

Each box represents one position. Replacing the displayed digit with nine alternatives gives 6 × 9 = 54 mutated integers.

The Complete 294001 Mutation Check

A full factorization is unnecessary when checking whether a result is composite. Finding one nontrivial divisor is enough. The expandable tables show all 54 mutations and one divisor for each result.

Why Digit Changes Often Produce Composite Numbers

The 54 results do not all become composite for the same reason. Some are caught by elementary divisibility rules. Others require less obvious divisors such as 173, 397, 409, or 587.

The units digit removes many candidates immediately

A multi-digit prime greater than 5 must end in 1, 3, 7, or 9. If its units digit changes to 0, 2, 4, 5, 6, or 8, the new number is automatically composite.

This explains six of the nine units-digit mutations of 294001. The remaining results ending in 3, 7, or 9 still need separate checks:

• 294003 is divisible by 3.
• 294007 is divisible by 7.
• 294009 is divisible by 3.

A convenient ending is not enough by itself. A digitally delicate prime must also fail every replacement in every earlier position.

Digit sums catch many mutations

A decimal integer is divisible by 3 when its digit sum is divisible by 3. Changing one digit changes that sum, so many mutations become easy to classify.

Consider the thousands digit:

The new digit sum is:

Since 12 is divisible by 3, 290001 is composite. The same rule catches results such as 194001, 494001, 224001, 293001, 294201, and 294081.

Other mutations need modular arithmetic

Divisibility by 2, 3, 5, or 9 explains only part of the pattern. A result such as 291001 has digit sum 13 and ends in 1, so the common tests do not settle it. It is still composite:

This mixture of easy and less obvious divisors is why delicate primes are more interesting than numbers designed around one simple divisibility trick.

The Place-Value Formula Behind a Digit Mutation

Let N contain the digit a in the position worth 10^k. Replacing that digit with b changes the number by:

For example, the hundreds digit of 294001 is 0. Replacing it with 1 adds 100:

The result factors as:

The formula gives a clean way to generate every mutation without treating the number as plain text. It also connects the topic to congruences: the powers 10^k repeat in predictable cycles modulo many primes. Mathematicians use those cycles to arrange divisibility across whole families of digit changes.

Digitally Delicate Primes as Prime-Free Neighbors

Two equal-length decimal strings are one-digit neighbors when they differ in exactly one position. In coding theory, the number of differing positions is called the Hamming distance.

A digitally delicate prime has no prime number at decimal Hamming distance 1, provided the comparison uses the standard written positions and allows the first digit to become zero.

This does not say anything about numbers two or more digit changes away. A delicate prime may have another prime nearby in ordinary numerical distance, or it may reach a prime after changing two digits. The definition controls only one replacement in one position.

How a Digitally Delicate Candidate Is Verified

A correct check has two parts. First, the original integer must be prime. Second, every allowed mutation must be composite.

Confirm the original number

Testing mutations is meaningless if the starting number is composite. The primality of the candidate comes first. The original integer and individual mutations can be checked with the Prime Number Checker.

Generate every distinct replacement

For each written position, replace the current digit with 0 through 9 while skipping the unchanged digit. A six-digit candidate produces 54 distinct checks. A ten-digit candidate produces 90.

Stop early when searching for a failure

Finding one prime mutation proves that a candidate is not digitally delicate. This makes rejection fast in many cases.

Confirmation works differently. To prove that a candidate is delicate, the test must reach the end without finding a single prime mutation.

Primality testing and factorization are different jobs

A program does not need the complete factorization of every mutation. It only needs a reliable proof that each result is composite. Trial division can expose small factors, while larger inputs may use stronger primality or compositeness tests.

For educational display, showing one divisor is useful. For computation, full factorization usually wastes time.

The First Digitally Delicate Primes

The decimal sequence begins:

There are no one-, two-, three-, four-, or five-digit decimal examples. The first appears at six digits.

How the Idea Entered Number Theory

The underlying problem appeared in 1978: could a prime become composite after every possible change to one decimal digit? Published solutions followed in 1979, including an argument that infinitely many such primes exist.

The name digitally delicate prime came later. Research also established that a positive proportion of primes, in an asymptotic sense, have the property in any fixed number base.

“Positive proportion” should not be read as “most primes.” It means the class does not fade to zero density among primes under the theorem’s counting method. The result does not supply a simple everyday percentage for a random finite interval.

The Number Base Changes the Property

Digital delicacy belongs to a number’s representation, not only to its numerical value. A number may be delicate in base 10 and fail in another base because the available digits and place values change.

In base B, each position has B − 1 alternative digits. An n-digit number therefore has:

one-digit mutations.

A careful statement should therefore say digitally delicate in base 10 unless the base is already clear.

Digitally Delicate and Widely Digitally Delicate

The ordinary definition considers the digits that appear in the written number. A stronger version imagines infinitely many zeros to the left:

A widely digitally delicate prime must produce a composite number after changing any existing digit or any one of those leading zeros.

Why 294001 fails the wider condition

Changing a leading zero to 1 at the ten-millions position gives:

10294001 is prime. Therefore, 294001 is digitally delicate but not widely digitally delicate.

How an infinite set of positions can be handled

Testing leading zeros one by one would never finish. Proofs use the repeating behavior of powers of 10 modulo selected primes.

A covering system divides all possible position exponents into finitely many residue classes. Each class is paired with a divisor, so a digit inserted at any covered position produces a composite number. The positions are infinite, but the modular patterns repeat.

The First Explicit Widely Digitally Delicate Prime

Existence was known before a concrete example could be displayed. The first published explicit example contains 4,030 decimal digits.

Its construction combined covering systems, the Chinese Remainder Theorem, a search through an arithmetic progression, mutation checks, and a separate primality proof. The search first produced a probable prime candidate. Its primality was then proved, and the divisibility conditions for the digit changes were verified.

The full 4,030-digit integer is not needed to understand the distinction. Its size mainly reflects how demanding explicit construction is, not a rule that widely delicate primes must always have thousands of digits.

Consecutive Delicate Primes

Digitally delicate primes do not have to appear as isolated members of the prime sequence. For every positive integer k, there exist k consecutive primes that are widely digitally delicate.

Here, “consecutive primes” means neighboring entries in the ordered list of primes. It does not mean consecutive integers.

The theorem shows that arbitrarily long runs can occur in which every prime loses primality after any allowed one-digit change, including a change to a leading zero.

Related Digit-Sensitivity Properties

Several nearby terms describe different operations. They should not be treated as synonyms.

Translated unit-sensitive primes

A 2026 number-theory preprint studies another related class called d-translated unit-sensitive primes. The operation appends d zeros to the right and then changes the new units digit. Only the final position is tested.

For example, when d = 1, the prime 97 leads to numbers from 970 through 979 after a units-digit choice. Since 971 and 977 are prime, 97 is not 1-translated unit sensitive.

This newer definition concerns a repeating units-digit condition. It is not a form of ordinary digital delicacy, which tests every existing position.

Common Definition Errors

Testing only one alternative per position

Changing each digit once is not enough. Every position has nine possible replacements in base 10.

Counting the unchanged number as a mutation

Replacing 4 with 4 changes nothing. It must be skipped. The total remains nine alternatives per position, not ten.

Calling every nonprime result composite

Zero and one require separate treatment because neither is composite.

Ignoring a leading zero created at the first position

Changing the first written digit to zero is allowed. The resulting shorter integer still needs a compositeness check.

Testing only the units digit

Units-digit replacements are often easy, but they represent only one position. A digitally delicate prime must also survive changes in the tens, hundreds, thousands, and all earlier positions.

Confusing ordinary and wide delicacy

Standard delicacy ignores extra leading zeros. Wide delicacy includes them. The fact that 294001 passes all 54 standard tests does not settle the wider condition.

Questions About Digitally Delicate Primes