How Can Infinitely Many Primes Be Infinitely Far Apart?

7 is made up of 7 by 7. We have a large gap.

It is easy to generalize. There is a sequence.

2, 3, and 4!

There is a prime gap with a length of at least n 1 when there is a sequence of n 1 consecutivecomposite numbers. Out along the list of natural numbers there are places where the closest primes are 100, or 1000, or even 1,000,000,000 numbers apart.

The results show a tension. Primes can be infinitely far apart. There are many primes that are close together. A decade ago the work of Yitang Zhang set off a race to close the gap and prove that there are infinitely many pairs of primes that differ by just 2. One of the most famous open questions in mathematics is the twin primes conjecture, and James Maynard has made his own significant contributions towards proving it.

Recent results about digitally delicate primes show this tension. Take a moment to ponder the question, "Is there a two-digit prime number that always becomes composite with any change to its digits?"

The number 23 is a good number to play around with. It is a prime, but what happens if you change its digit? All except 20, 22, 24, 26 and 28 are even. So far, it's been great. You get 29 if you change the digit to a 9. We are not looking for a prime like 23.

Do you think about 37? We don't need to check numbers that end in 5, so we'll just check 31, 33 and 39 37 doesn't work since 31 isprime.

Is it possible that a number like that exists? The answer is yes, but we have to go all the way up to 97 in order to find it.

If you change any of the digits of a prime number to something else, it will lose its primordiality. We have seen that 97 is delicate in the digit, but does it meet the full criteria of being digitally delicate? If you change the digits to 1 you get a prime. 37,47 and 67 are all primes.

There is no two-digit prime. The table below shows why.

All the numbers in a row have the same digit and all the numbers in a column have the same digit. The fact that 97 is the only shaded number in the row shows that it is not delicate in the digits.

The only prime in the row and column is a two-digit one. There is no two-digit prime shown in the table. There is a three-digit prime. The layout of the three-digit primes between 100 and 199 is shown in this table.

It is delicate in the digits because it is in its own row. Primes can be produced by changing the tens digit to 0 for 103 or 6 for 163. There is no three-digit number that is guaranteed to be interchangeable if you change its digits. There can't be a three-digit prime. We didn't check the number. A three-digit number has to avoid primes in three directions in a three-dimensional table.

Is there a digitally delicate prime? The primes are less likely to cross paths in the rows and columns when you go further out on the number line. A prime is less likely to be digitally delicate if there are more digits in the number.

You will discover that digitally delicate primes exist if you continue. The smallest is more than two hundred thousand dollars. The number you get when you change one of the digits will be a mixture. There are more to come: 594,171, 971,767, and 1,062,599. They don't stop There are many delicate primes, according to the mathematician Paul Erds. That was the first result of the numbers.

Erds showed that there are infinitely many digitally delicate primes in any base. If you choose to represent your numbers in a different way, you will still find many delicate primes.

Digitally delicate primes comprise a non zero percentage of all prime numbers. The number of digitally delicate primes to the number of primes overall is greater than zero. There is a positive proportion of all primes that are digitally delicate. Since there are fewer and fewer primes the farther out you go, the primes themselves don't make up a positive proportion. The ratio of delicate primes to total primes is kept above zero by digitally delicate primes.

A new variation of these strange numbers was discovered in 2020. Instead of thinking about 97 by itself, mathematicians thought of it as having leading zeros, by relaxing the concept of what a digit is.

0000000097 is the number

The question of digital delicacy can be extended to these new representations. If you change any of the digits, any of the leading zeros, and there's a widely digitally delicate prime, you'll get a composite prime. We know that the answer is yes thanks to the work of the mathematicians. There are a lot of delicate primes.

Professionals and enthusiasts can play with prime numbers. You can count on mathematicians to keep discovering and inventing new kinds of primes.

Exercises

This is the first thing. The biggest prime gap is between 2 and 101.

There are two To prove that there are infinitely many primes, it is necessary to assume that there are finitely many primes. Doesn't this mean that q needs to be the best?

There are three. There is a prime between k and 2k. It is difficult to prove that there is a prime between k and q. Let us know if it's true.

There are four. Is it possible to find the smallest prime number in the ones and tens digits. Changing the digits will produce a number. You could write a computer program to do this.

Can you find the smallest prime number when it's represented in a pair of numbers? The power of 2 is represented by the digits 0 and 1 in base 2.

Click for the first answer.

The largest gap is between the primes 89 and 97. Generally speaking, the gaps get larger as you go further out along the number line, but of course the twin primes conjecture claims that there will always be primes very close together no matter how far out you go. Notice also how inefficient the method for constructing prime gaps used in this column is: To construct a prime gap of this size, you would start with the number 8!+2=40,322 .

You can click for answer 2.

It is not possible to say yes. Consider the first six primes. The number q is 2 3 5 7 11 13 This is not a prime, but it is a factor. Prime factors are larger than the first six.

Click for the third answer.

We're done if either k or q isprime. We already know that it is not divisible by any of the first n primes. Since these are all primes less than k, the prime must be larger than k. There must be a prime between k and q in order for it to be less than q.

Click for the correct answer.

It fails once you consider the hundreds digit because 2,459 is a prime. Thanks to the mathematician John D. Cook.

Click to find an answer to the problem.

Since 126, 125, 123, and 121 are all related to the same number, it's a bit delicate.