Diophantine Equations

Today I turned 24521 days old and it turns out that this number is a non-negative value of x in the solution (x, y) to the Diophantine equation: 

x^2+(x+16807)^2=y^2

The corresponding y value is 48055 so the solution is (24521, 48055). So I thought that this would be an appropriate time to include some information about Diophantine equations. According to Wikipedia:

The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.

 The definition given in the same article is:

In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An exponential Diophantine equation is one in which exponents on terms can be unknowns.

Today's Diophantine equation is quadratic and it's equivalent to finding integer solutions to the right-angled triangle with hypotenuse y and arms x and x + 16807. As it turns out, the requisite triangle is 48055, 24521 and 41328. In general terms, finding all right triangles with integer side-lengths is equivalent to solving the Diophantine equation a^2 + b^2 = c^2, except in this case we know that b = a + 16807.

Diophantine analysis is a big topic to which I may return at a later date but that's enough for now.

Primes Within Primes

Today is a prime day, 24517, and it turns out to be an interesting prime in that it conforms to the membership requirements of OEIS A164077 which state that it must be:

• Prime p1 of the form a^b + c^d = p1, where a, b, c, d are primes
• a + b + c + d = p2, where p2 (A164078) is prime
• conc(abcd) = p3 (concatenation of a, b, c , d) is prime (A164079)

The first such number to qualify is 5527 because:

• 5^5 + 2^7 = 3253 is prime
• 5 + 5 + 2 + 7 = 19 is prime
• conc (abcd) = 5527 is prime

The next number to qualify is 24517 because:

• 29^3 + 2^7 = 24517 is prime
• 29 + 3 + 2 + 7 = 41 is prime
• conc (abcd) = 29327 is prime

After that comes 78157 because:

• 2^5 + 5^7 = 78157 is prime
• 2 + 5 + 5 + 7 = 19 is prime
• conc (abcd) = 2557 is prime

After that comes 366103 because:

• 2^13 + 71^3 = 366103 is prime
• 2 + 13 + 71 + 3 = 89 is prime
• conc (abcd) = 213713 is prime

... and so on and so on ...

Clearly primes satisfying such stringent prime number requirements are few and far between. Here is the list as shown on the OEIS site:

3253, 24517, 78157, 366103, 548677, 705097, 1030429, 1229257, 5735467, 6438391, 12221371, 17498881, 19618243, 74084347, 118370899, 263374849, 270840151, 286199371, 410180599, 418195621, 418719781, 529483321, 565609411, 698388391

I would imagine that there are an infinity of such primes but don't ask me to prove it.

Numbers That Are Sums of Two Squares

Today I turned 24505 days old and was surprised to find the following information about this number:

In all the time that I've been analysing my daily numbers, this was the first time that I'd seen the number being representable by the sum of the squares of six different pairs of numbers. Similarly, I'd not seen a number being the hypotenuse of four different Pythagorean triples. It lead me to think whether this was a record of some sort.

Well, a little research shows that this is certainly not the case. For example, the first numbers that are representable as the sum of the squares of six different pairs of numbers are:

5525, 9425, 11050, 12025, 12325, 13325, 14365, 15725, 17225, 17425, 18785, 18850, 19825, 21125, 22100, 22525, 23725, 24050, 24505, 24650, 25925, 26650, 26825, 27625, 28730, 28925, 29725, 31025, 31265, 31450, 31525, 32045, 32825, 34450, 34645, 34850

There are numbers that are representable as the sum of the squares of seven different pairs of numbers but the first such number is 105625. Numbers that are representable as the sum of the squares of eight different pairs of numbers are interestingly far more common than seven and begin thus:

27625, 32045, 40885, 45305, 47125, 55250, 58565, 60125, 61625, 64090, 66625, 67405, 69745, 77285, 78625, 80665, 81770, 86125, 87125, 90610, 91205, 94250, 98345, 98605, 99125, 99905, 101065, 107185, 110500, 111605, 112625, 114985, 117130, 118625

71825, 93925 and 122525 are the first three numbers that are representable as the sum of the squares of nine different pairs of numbers. Of numbers that are representable as the sum of the squares of ten different pairs of numbers, the first is 138125. The OEIS doesn't give any results for 11 onwards but there's no reason to not suppose that numbers exist that are representable as the sum of the squares of eleven different pairs of numbers and beyond.

Let's summarise the results:

1. 5
2. 65
3. 325
4. 1105
5. 8125
6. 5525
7. 105625
8. 27625
9. 71825
10. 138125

I won't deal here with the numbers n for which n^2 can be represented as the sum of squares of different pairs of numbers, in other words n forms the hypotenuse of a Pythagorean triple. However, a similar analysis is possible.

Achilles Numbers, Powerful Numbers and Perfect Powers

Today I turned 24500 days old and an analysis of this number revealed that it was an Achilles number in the OEIS listings, specifically OEIS A203663: Achilles number whose double is also an Achilles number. The initial members of this sequence are listed as:

432, 972, 1944, 2000, 2700, 3456, 4500, 5292, 5400, 5488, 8748, 9000, 10584, 10800, 12348, 12500, 13068, 15552, 16000, 17496, 18000, 18252, 21168, 21296, 21600, 24300, 24500, 24696, 25000, 26136 

I'd not heard of an Achilles number before and so I investigated. Wikipedia provides this definition:

An Achilles number is a number that is powerful but not a perfect power. A positive integer \(n\) is a powerful number if, for every prime factor \(p\) of \(n\), \(p^2\) is also a divisor. In other words, every prime factor appears at least squared in the factorisation. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as \(m^k\), where \(m\) and \(k\) are positive integers greater than 1.

Now 24500 factorises to \(2^2 \times 5^3 \times 7^2\) and is thus powerful but not a perfect power. So it is an Achilles number. Multiplication by 2 does not change this because 49000 factorises to \(2^3 \times 5^3 \times 7^2\) and again powerful but not a perfect power.

Here is the Wikipedia definition of a perfect power:

A perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, \(n\) is a perfect power if there exist natural numbers \(m > 1\), and \(k > 1\) such that \(m^k = n\). In this case, \(n\) may be called a perfect \(k\)-th power. If \(k = 2\) or \(k = 3\), then \(n\) is called a perfect square or perfect cube, respectively. Sometimes 1 is also considered a perfect power (\(1^k = 1\) for any \(k\)).