Idoneal Numbers

Euler's numeri idonei or idoneal numbers (suitable or convenient numbers) were included in four papers that Euler presented to the Petersberg Academy in 1778. They are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848

The property that these numbers share is that they cannot be represented in the form \( ab+ac+bc \) with \(0 < a < b < c\). While 1848 is the largest known number, this source states that:

S. Chowla proved (in 1934) that the number of numeri idonei is finite, and it is known that there can be at most ONE more square-free numerus idoneus beyond those found by Euler.  Whether such another number exists is still an open question.

However, another source states that 1848 is the largest number if the Riemann Hypothesis hold true otherwise what's said above will hold.

I came across the reference to these numbers when investigating the number associated with my diurnal age today, namely 27358. It is a member of OEIS

A094377

Greatest number having exactly n representations as \(ab+ac+bc \) with \(0 < a < b < c\).

Up to 40,000, the initial members of this sequence are:

1848, 193, 1012, 862, 3040, 2062, 4048, 3217, 7392, 4162, 7837, 8002, 12397, 13297, 14722, 16417, 21253, 21058, 30493, 27358, 34357, 34318

Here we see 1848 appearing as the greatest number for the case of \(n=0\), in other words it cannot be represented in this way. On the other hand, 193 is the greatest number that can be represented in only one way, namely:$$193=4 \times 7 + 4 \times 15 + 7 \times 15\\ \text{where } a=4, b=7 \text{ and } c=15$$27358 corresponds to the case of \(n=19\), in other words this number can be represented in 19 different ways in the form \(ab+ac+bc \). Here are 18 of them with one missing that I haven't been able to find.

• 5 - 134 - 192
• 14 - 32 - 585
• 19 - 34 - 504
• 23 - 56 - 330
• 26 - 81 - 236
• 26 - 105 - 188
• 29 - 134 - 144
• 30 - 41 - 368
• 30 - 112 - 169
• 34 - 72 - 235
• 35 - 66 - 248
• 44 - 53 - 258
• 44 - 107 - 150
• 56 - 102 - 137
• 57 - 70 - 184
• 66 - 91 - 136
• 71 - 108 - 110
• 9 - 14 - 1184

There's a lot more to this topic but let's return to the idoneal numbers and find out why they are "convenient". Well, they are convenient because they were used historically to help find large primes using the formula:$$x^2+n\, y^2\\ \text{ where } n \text{ is a convenient number}$$For example, Euler was able to find the prime:$$18,518,809=197+1848 \times 100$$where as can be seen the largest convenient number makes its appearance. If we replace 197 with other primes, we find that in the prime range up to 600, 46.3% of the resultant numbers are prime (permalink). It would be interesting to see how this figure compares to others using different convenient numbers and/or different values of \(x\) and \(y\).

This is a fairly deep topic and I won't go further into it here but it's clear that the idoneal numbers are a small but fascinating group.

Congruent Numbers

Over the years I've sometimes made note of when a number associated with my diurnal age is congruent. Often I've just ignored the fact. The definition of such number is:

A number is called congruent it is the area of a right triangle with rational sides.

The first congruent number is 5 because it is the area of a right triangle with sides of 20/3, 3/2, and 41/6. The next number, 6, is also congruent being the area of the famous 3, 4, 5 right triangle. See Figure 1.

Figure 1

7 is also a congruent number being the area of a right triangle with sides such that (see Figure 2):$$ \begin{align} \Big ( \frac{35}{12} \Big )^2+ \Big ( \frac{24}{5} \Big )^2 &= \Big ( \frac{337}{60} \Big )^2 \\ \frac{1}{2} \cdot \frac{35}{12} \cdot \frac{24}{5} &= 7 \end{align}$$

Figure 2

With 7, we begin to see the problem of determining whether a number is congruent or not. How do we find those fractions that confirm that 7 is congruent. It's not easy. To quote from Wikipedia:

The question of determining whether a given rational number is a congruent number is called the congruent number problem. This problem has not (as of 2019) been brought to a successful resolution. Tunnell's theorem provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture, which is still unproven.

Without getting too deeply into this topic, it suffices to say that OEIS A006991 provides a list of so-called primitive congruent numbers up to 10,000.  These are congruent numbers that are square-free. If any of these numbers are multiplied by a positive square number, a new congruent number is created.

Let's take the number associated with my diurnal age yesterday as an example. 27260 factorises as follows:$$ \begin{align} 27260 &= 2^2 \times 5 \times 29 \times 47\\ &= 2^2 \times 6815 \end{align}$$Now 6815 is a member of OEIS A006991 and therefore 27260 is thus a congruent number because a primitive congruent number, 6815, has been multiplied by a positive square number (4).

It can be noted that 6814 is also a member of OEIS A006991 and thus 4 x 6814 = 27256 is also a congruent number. It's not uncommon for primitive congruent numbers to occur in pairs or even triples.  For example: (5, 6, 7), (13, 14, 15), (21, 22, 23), (29, 30, 31), 34 is a singleton, (37, 38, 39) etc.

What's Special About 26962?

I'm following my tradition of making a special post for my palindromic days. The last one was 26862 and titled What's Special About 26862? Now I've reached 26962 and this palindrome has some interesting properties. The first is that it can be split up into two factors, each of which is the reverse of the other (with square numbers such as 121 excluded).

26962 = 122 x 221 = 221 x 122

There are not many palindromes with this property in the range up to one million. Here is the list:

• 252 =12 x 21 = 21 x 12
• 20502 = 102 x 201 = 201 x 102
• 23632 = 112 x 211 = 211 x 112
• 26962 = 122 x 221 = 221 x 122

As can be seen, 26962 is last palindrome with this property in the range up to one million. We have to extend the range in order to find more. In the range between one and two million, the following palindromes are found (permalink):

• 1113111 = 1011 x 1101 = 1101 x 1011
• 1226221 = 1021 x 1201 = 1201 x 1021
• 1357531 = 1121 x 1211 = 1211 x 1121

Notice how all the factors are comprised of the digits 0, 1 and 2 only. If we relax the rule that square numbers are excluded then, with their inclusion, 26962 is a member of OEIS A158642:

A158642

Palindromic numbers which are the product of a number n and its reversal (n written backwards)

The members of this sequence, up to two million, are:

0, 1, 4, 9, 121, 252, 484, 10201, 12321, 14641, 20502, 23632, 26962, 40804, 44944, 1002001, 1113111, 1226221, 1234321, 1357531

The previously mentioned non-square numbers are marked in blue.

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26962 is a palindrome which is 

a product of a pair of emirpimes

In the case of 26962, it can be noted that the factors 122 and 221 are products of emirpimes:

• 122 = 2 x 61
• 221 = 13 x 17

This property of being a product of emirpimes qualifies 26962 for membership in OEIS A158126:

A158126

Products of emirpimes pairs, sorted.

The initial members of the sequence are:

765, 1612, 3627, 4606, 4930, 26962, 39483, 48763, 58765, 61306, 69723, 85405, 102910, 107485, 118809, 129682, 134458, 136467, 140572, 146047, 148930, 151209, 155038, 162409, 178555, 194242, 196315, 203098, 213310, 236421, 283798, 291247

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26962 is a palindrome 
with four distinct prime factors

26962 has four distinct prime factors: 2, 13, 17 and 61. Palindromes with this property are fairly rare and constitute OEIS A046394: 

A046394

Palindromes with exactly four distinct prime factors.

The initial members of this sequence, up to 100,000, are (permalink):

858, 2002, 2442, 3003, 4774, 5005, 5115, 6666, 10101, 15351, 17871, 22422, 22722, 24242, 26562, 26962, 28482, 35853, 36363, 41314, 43734, 43834, 45654, 47874, 49494, 49794, 49894, 51015, 51315, 51415, 53535, 53835, 53935, 56865, 58485, 59295, 59595, 60006, 62526, 62826, 64246, 64446, 66666, 66766, 68286, 73437, 74347, 78387, 81618, 81718, 83638, 83838, 87078, 87178, 89598, 89698, 92829, 96369, 98889

26962 is also a palindrome in base 11, being represented as 19291. This property qualifies it for membership in OEIS A180454.

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26962 can be written as a sum of two distinct palindromic primes in three different ways

Still on the topic of palindromes, 26962 is a member of OEIS A356854:

A356854

Palindromes that can be written in more than one way as the sum of two distinct palindromic primes.

The primes are 10301 + 16661 = 10601 + 16361 = 11411 + 15551 = 26962.

The initial members of the sequence are (permalink):

282, 484, 858, 888, 21912, 22722, 23832, 24642, 25752, 26662, 26762, 26862, 26962, 27672, 27772, 27872, 27972, 28482, 28782, 28882, 28982, 29692, 29792, 29892, 29992, 40704, 41514, 41614, 41814, 42624, 42824, 42924, 43434, 43734, 43834, 43934, 44744, 44844, 44944, 45354

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26962 is a palindromic Ulam number

26962 is also an Ulam number and so, being a palindromic Ulam number, it qualifies for membership in OEIS A173542:

A173542

Palindromic Ulam numbers.

The initial members of the sequence are:

1, 2, 3, 4, 6, 8, 11, 77, 99, 131, 282, 363, 414, 434, 585, 646, 949, 2112, 2332, 2552, 2662, 5335, 5665, 8008, 8338, 8668, 10501, 13531, 13931, 15251, 16961, 17071, 18381, 18581, 18681, 22122, 22322, 23632, 23932, 25452, 26962, 28582, 28682, 30703, 30803, 32123

Primitive Practical Numbers

On July 31st 2018, three years and one day ago, I created a post titled Practical Numbers. To quote from that post:

In number theory, a practical number or panarithmic number is a positive integer \(n\) such that all smaller positive integers can be represented as sums of distinct divisors of \(n\). For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2. Wikipedia.

That is straightforward enough and, before we progress to the concept of primitive practical numbers, let's familiarise ourselves with the concept of a complete sequence. Here is a definition:

In mathematics, a sequence of natural numbers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.

For example, the sequence of powers of two (1, 2, 4, 8, ...), the basis of the binary numeral system, is a complete sequence; given any natural number, we can choose the values corresponding to the 1 bits in its binary representation and sum them to obtain that number (e.g. \(37 = 100101_2 = 1 + 4 + 32\)). This sequence is minimal, since no value can be removed from it without making some natural numbers impossible to represent. Simple examples of sequences that are not complete include the even numbers, since adding even numbers produces only even numbers—no odd number can be formed. Wikipedia.

Primitive practical numbers are defined in OEIS A267124:

A267124

Primitive practical numbers: practical numbers that are square-free or practical numbers that when divided by any of its prime factors whose factorisation exponent is greater than 1 is no longer practical.

 The accompanying comments in the OEIS entry are informative:

If \(n\) is a practical number and \(d\) is any of its divisors then \(n \times d\) must be practical. Consequently, the sequence of all practical numbers must contain members that are either square-free (A265501) or when divided by any of its prime factors whose factorisation exponent is greater than 1 is no longer practical. Such practical numbers are said to be primitive. The set of all practical numbers can be generated from the set of primitive practical numbers by multiplying these primitives by any of their divisors.

The examples are given of:

• \(a(4)=20=2^2 \times 5\). It is a practical number because it has 6 divisors 1, 2, 4, 5, 10, 20 that form a complete sequence. If it is divided by 2 the resultant has 4 divisors 1, 2, 5, 10 that is not a complete sequence. Thus it is a primitive practical number.
  
  
• \(a(7)=42=2 \times 3 \times 7\). It is square-free and is practical because it has 8 divisors 1, 2, 3, 6, 7, 14, 21, 42 that form a complete sequence. Thus it is a primitive practical number.

12 as an example of a practical number that is not primitive. It has divisors 1, 2, 3, 4, 6, 12 that form a complete sequence. It is not square-free and, when divided by 2 (a prime factor whose factorisation exponent is greater than 1), the result is 6 which is still a practical number. 6 in fact is a primitive practical number because it is a practical number that is square-free.

The initial primitive practical numbers are:

1, 2, 6, 20, 28, 30, 42, 66, 78, 88, 104, 140, 204, 210, 220, 228, 260, 272, 276, 304, 306, 308, 330, 340, 342, 348, 364, 368, 380, 390, 414, 460, 462, 464, 476, 496, 510, 522, 532, 546, 558, 570, 580, 620, 644, 666, 690, 714, 740, 744, 798, 812, 820, 858, 860, 868, 870, 888, 930, 966, 984 

Let's examine the first 8 primitive practical numbers: 1, 2, 6, 20, 28, 30, 42 and 66. The differences between these terms are 1, 4, 14, 8, 2, 12 and 24. The record gaps are 1, 4, 14 and 24, which occur after the terms 1, 2, 6 and 42. If we extend our examination of these record gaps and note the terms before which they occur, we end up with OEIS A334883:

A334883

Primitive practical numbers (A267124) with a record gap to the next primitive practical number.

The initial terms making up this sequence are:

1, 2, 6, 42, 104, 140, 1036, 1590, 2730, 7900, 10374, 19180, 22660, 23180, 26418, 105868, 114960, 139060, 295780, 403524, 482250, 1294144, 2468944, 4799058, 5379282, 19035500, 20233936, 21803860, 112406992, 789190976, 3520928922

What drew my attention to this sequence and to primitive practical numbers in general was the fact that my diurnal age today is 26418. The gap to the next primitive practical number 26572 is 154 but that record is not broken until 105868, after which there is a gap of 188 to 106056.