Deletion of Digits from any side of a Number – Forming a Prime

Finally, a study of mathematics and its contributions to the sciences exposes a deep question. Mathematics is man-made. The concepts, the broad ideas, the logical standards and methods of reasoning, and the ideals which have been steadfastly pursued for over two thousand years were fashioned by human beings. Yet with this product of his fallible mind man has surveyed spaces too vast for his imagination to encompass; he has predicted and shown how to control radio waves which none of our senses can perceive; and he has discovered particles too small to be seen with the most powerful microscope. Cold symbols and formulas completely at the disposition of man have enabled him to secure a portentous grip on the universe. Some explanation of this marvelous power is called for.

Morris Kline

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Prime numbers are an integral part of mathematics. A prime number is always divisible by 1 and the number itself. Hence, it has only two positive factors. Starting from a prime number, if we successively delete the extreme left or right or both digits, we will again obtain a prime! In this blog post, we will consider these prime numbers, known as truncatable primes.

Automatically, the question that comes to our mind is, what is a truncatable prime? A number is a truncatable prime if successively removing digits (until only one or two digits remain) forms another prime number.

Removal can be done from both sides—right and left. From which sides should we remove the digits? If these digits are removed from the left side, the number is a left-truncatable prime; if they’re removed from the right side, the number is right-truncatable. More formally, we have the following:

Left Truncatable Prime

A left truncatable prime is a prime number which, in a given base,

• (i) contains no 0;
• (ii) if the leading left digit is successively removed, then all resulting numbers are prime.

Consider, for example, 9137.

9137 is a prime

137 is a prime

37 is a prime

7 is a prime

In base 10, there are exactly 4260 left-truncatable primes. The largest known left-truncatable prime is the 24-digited number 357686312646216567629137.

Right Truncatable Prime

A right truncatable prime is a prime number which, in a given base,

• (i) contains no 0;
• (ii) if the leading right digit is successively removed, then all resulting numbers are prime.

Consider, for example, 7393.

7393 is a prime

739 is a prime

79 is a prime

7 is a prime

In base 10, there are exactly 83 right-truncatable primes. The largest known right-truncatable prime is the 8-digited number 73939133.

Left and Right Truncatable Prime

A left and right truncatable prime is a prime number which, in a given base,

• (i) contains no 0;
• (ii) if the leading left and right digits are removed together, then all resulting numbers are prime. This can be done until the last number becomes a one– or two–digit prime.

Consider, for example, 1825711.

1825711 is a prime

82571 is a prime

257 is a prime

5 is a prime

In base 10, there are exactly 920720315 left-and-right-truncatable primes. Among them 331780864 are left and right truncatable primes with an odd number of digits. The largest one is the 97-digited number 7228828176786792552781668926755667258635743361825711373791931117197999133917737137399993737111177.

There are 588939451 left and right truncatable primes with an even number of digits. The largest is the 104-digited prime number 91617596742869619884432721391145374777686825634291523771171391111313737919133977331737137933773713713973.

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