Beyond the ordinary
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Beyond The Ordinary
The number 24019 stands out primarily as a mathematical anomaly and a geographical anchor.
Mathematical Properties.
It is a Prime Number:
24019 is a prime number, meaning it is only divisible by 1 and itself.
Permutable Pair:
The digits of 24019 can be rearranged to form other primes, such as 10249, 10429, 14029, 20419, and 42019.
The original search I performed on the primality of 24019, supplied the following information block:
Emordnilap / Semordnilap Prime: If you reverse its digits, it becomes 91042, which is also a prime number.
Strictly (mathematically) this is incorrect; but, since 91042 is 910+42, which is 910+Ꝝ = 910Ꝝ, maybe there's some truth there.
Geographic & Everyday Significance.
ZIP Code: In the United States, 24019 is the ZIP code for part of Roanoke, Virginia, a major hub nestled in the Blue Ridge Mountains.
The digits of 24019 can be rearranged to form other primes, such as 10249, 10429, 14029, 20419, and 42019. Is 24019 unusual for this.
Are there any more prime permutations of 24019?
24019 is not a permutable prime. While 24019 itself is a prime number, a permutable prime (also known as an anagrammatic or absolute prime) must remain prime after every possible rearrangement of its digits.
Multi-digit permutable primes can only contain the digits 1, 3, 7, and 9. They cannot contain any even digits (0, 2, 4, 6, 8) or the digit 5, because rearranging those numbers to the ones place would make the new number divisible by 2 or 5.
Because 24019 contains the digits 2, 4, and 0, you can easily permute its digits into composite numbers like 14902 (which is even) or 12490 (which is a multiple of 10).
24019 is not unusual for having a lot of permutation primes. Rearranging its digits yields 16 different 5-digit prime numbers, which is slightly below average for a 5-digit prime with five distinct digits.
An analysis of 5-digit primes with entirely unique digits.
The Average:
A typical set of 5 distinct digits that can form at least one prime will usually yield about 22 different prime permutations.
The Record Holders:
Some sets of 5 distinct digits can form up to 39 different prime numbers (for example, permutations of the digits in 12379).
The full list of the 16 prime permutations for the digits {0, 1, 2, 4, 9} is:
10429, 12049, 12409, 14029, 20149, 24019, 24091, 24109, 29401, 40129, 42019, 42901, 49201, 92041, 92401, and 94201 (excluding permutations starting with a leading zero).
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5
The absolute 5-digit record holder is the digit set {1, 3, 7, 8, 9}, which can be rearranged to form 39 different prime numbers. Out of 120 total possible mathematical permutations, nearly one-third of them result in a prime number.
The full list of these 39 record-breaking primes is:
13789, 13879, 17389, 17839, 18379, 18397, 18793, 18973, 19387, 37189, 38197, 38791, 38917, 38971, 71389, 71983, 73189, 73819, 78139, 78193, 79813, 81937, 81973, 83719, 83791, 87931, 89137, 89317, 89371, 91387, 91837, 91873, 93187, 93871, 97381, 97813, 98317, 98713, and 98731.
Why does this set do so well?
No Evens or Fives at the End:
It doesn't contain a 0, 2, 4, 5, or 6. This means no matter how you arrange them, the final digit can always be a 1, 3, 7, or 9, giving every single permutation a mathematical chance at being prime.
Digital Sum:
The sum of the digits is 28. Because 28 is not divisible by 3, none of these permutations will ever be divisible by 3, avoiding an automatic disqualification.
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6
The absolute 6-digit record-holding digit set is {1, 2, 3, 4, 7, 9}, which can be rearranged to form 148 different prime numbers.
Out of 720 total mathematical permutations possible with these 6 distinct digits, over 20% of them result in a prime number.
Why this specific set works so well.
Favorable Digital Sum:
The sum of these digits is 26 (1+2+3+4+7+9). Because 26 is not divisible by 3, none of its 720 permutations are divisible by 3, avoiding an automatic disqualification.
Low count of "bad" ending digits:
Only two digits in the set (2 and 4) make a number automatically even if they end up in the last position. The remaining four digits (1, 3, 7, and 9) all leave the door open for a number to be prime.
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7
The absolute 7-digit record-holding digit set is {1, 2, 3, 5, 7, 8, 9}, which can be rearranged to form an incredible 731 different prime numbers.
Out of 5,040 total mathematical permutations possible with these 7 distinct digits, roughly 14.5% of them are prime.
Why this set dominates.
Favorable Digital Sum: The sum of these digits is 35. Since 35 is not divisible by 3, none of its 5,040 permutations can be divided by 3.
Maximising "Good" Endings: Even though it contains a 2, 5, and 8 (which block a prime if they land in the final position), the set includes the ultimate prime-ending digits: 1, 3, 7, and 9.
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8
The absolute 8-digit record-holding digit set is {1, 2, 3, 4, 5, 6, 7, 9}, which can be rearranged to form an astonishing 4,333 different prime numbers.
Out of 40,320 total mathematical permutations possible with these 8 distinct digits, roughly 10.7% of them result in a prime number.
Why this set dominates.
Mathematical Eligibility: The sum of these digits is 37. Because 37 is not divisible by 3, none of its 40,320 permutations can be divided by 3, saving the entire set from automatic disqualification. (For comparison, picking the set {2, 3, 4, 5, 6, 7, 8, 9} sums to 44, which is also not divisible by 3, but it yields "only" 3,098 primes).
Optimised Endings: It completely excludes 0 and 8. By leaving out these heavy even digits, it increases the statistical probability of the remaining numbers ending in a valid prime-ending digit (1, 3, 7, or 9).
We have reached the point where the lists are thousands of numbers long!
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9
The 9-digit record holder is {0, 1, 2, 3, 4, 5, 6, 7, 9}, yielding a total of 26,519 prime numbers across its permutations.
To break this down further, out of the 362,880 total mathematical permutations possible with these 9 digits, roughly 7.3% of them result in a prime number.
Why This Specific Set Is Unique.
The Only Logical Group: Because of the divisibility rule of 3, any 9-digit combination that leaves out 0, 3, 6, or 9 will always have a digital sum divisible by 3. This automatically disqualifies the entire group from containing a single prime number.
The "8" Disadvantage: Leaving out 8 is statistically superior to leaving out 1, 2, 4, 5, or 7. By omitting 8, you eliminate a heavy even digit that would otherwise ruin a potential prime if it landed in the final slot.