Draft:Mortar numbers

A Mortar number n is an integer divisible by every integer up to the floor of its square root.

Put simply, if you take the square root of a Mortar number and round down, every integer up to that value evenly divides the number.

List of all known Mortar numbers:

list: 1, 2, 3, 4, 6, 8, 12, 24.

Example:

• 24 qualifies as a Mortar number because it is divisible by 1,2,3,4 (all integers up to √24≈4.89).
• 36 doesn't because it is divisibile by 1,2,3,4,6 but not 5.

They were discovered while programming an algorithm to find prime numbers recursively. A bug switched a simple condition: instead of filtering the number N² when not multiple of [1,2,3... N] they were filtered only when the previous condition was met.

It has been proven by the reddit user ElusiveMoose314 under the post of a user NotGioseaxMC inside the math subreddit that the largest Mortar number is 24.

Reddit user's pseudo proof:

Original proof can be found here.

If N is a Mortar number, then N must be divisible by all the integers less than or equal to the square root of N. Let n be the floor of the square root of N, meaning N needs to be divisible by all the integers 1, 2, 3, up to n. Consequently, N is divisible by the least common multiple (LCM) of these integers.

The LCM of 1, 2, 3, up to n can be considered similar to the product 1 * 2 * 3, and so on, up to n, but it avoids counting duplicate factors. For instance, the LCM of 1, 2, 3, 4, 5 is 1 * 3 * 4 * 5 because the factor 2 is already included in 4.

Interestingly, this LCM can be expressed as e raised to the power of psi(n), where psi is the second Chebyshev function. The function psi(x) is defined as the sum of the logarithms of the prime numbers up to x. For example, psi(5) is the sum of the logarithms of 2, 3, and 5. Careful consideration of the properties of this sum and exponent rules confirms that the LCM of 1, 2, 3, up to n equals e raised to the power of psi(n).

Using this formulation is beneficial because extensive research has been dedicated to establishing accurate upper and lower bounds for functions like the Chebyshev functions, which are crucial in analytic number theory and related to the Riemann hypothesis. Although the detailed derivation of these bounds is complex, the key insight is that psi(x) is approximately equal to x.

Assuming for simplicity that psi(x) equals x:

• We have N approximately equal to n squared because n is the floor of the square root of N.
• N must be at least as large as the LCM of 1, 2, 3, up to n, so N is greater than or equal to e raised to the power of n.

Since e raised to the power of n grows much faster than n squared, these two conditions are incompatible for large n.

To rigorously demonstrate this, we can use a specific lower bound for psi(x) instead of assuming psi(x) is approximately equal to x. For instance, using Nagura's bound:

psi(x) is greater than 0.916 x -2.318

We find that for n greater than or equal to 8, e raised to the power of 0.916 * n - 2.318 grows too large compared to (n + 1) squared. Therefore, beyond n equals 8, the conditions for being a Mortar number cannot be satisfied.

Since we know that up to 1000 the largest Mortar number is 24; it is the largest Mortar number.