Fermat Near Misses Revisited

On the 10th of December 2022, almost two years ago now, I made a post titled Fermat Near Misses where I wrote:

I was surprised to find that my diurnal age today, 26914, is part of a triple of numbers that almost satisfies the equation \(x^3+y^3=z^3\) and hence the term "Fermat near misses". There are of course no positive integer solutions to this equation but some solutions only miss out by 1.

For example, the triplet of numbers 9, 10 and 12 satisfy \(x^3+y^3=z^3+1\) where \(x=9\), \(y=10\) and \(z=12\):$$9^3+10^3=12^3+1=1729$$Here 1729 is the famous "taxi cab number". Today, the number associated with my diurnal age is 27630 and it has a property that qualifies it for membership in OEIS A050789:

A050789: consider the Diophantine equation \(x^3+y^3=z^3-1\) or 'Fermat near misses'. The values of \(z\) (see A050787) are arranged in monotonically increasing order. Sequence gives values of \(y\).

So we have:$$17328^3 +27630^3 = 29737^3-1 = 26296107018552$$The sequence begins as follows:

8, 138, 138, 426, 486, 720, 823, 812, 1207, 2292, 2820, 3230, 5610, 5984, 6702, 8675, 11646, 11903, 16806, 17328, 21588, 24965, 27630, 36840, 31212, 37887, 33857, 34566, 49409, 46212, 59022, 66198, 66167, 56503, 69479, 64165, 78244, 89970

Notice how 17328 reappears again, not as an \(x\) value, but as a \(y\) value:$$10866^3+ 17328^3 =18649^3-1 = 6485846213448$$Figure 1 shows the values of \(x\), \(y\) and \(z\) that satisfy these near misses:

Figure 1: permalink

Here is a sorted list of all the initial integers that are involved in these near misses, in other words all the \(x\)'s, \(y\)'s and \(z\)'s together.

6, 8, 9, 71, 135, 138, 144, 172, 236, 242, 372, 426, 486, 505, 566, 575, 577, 720, 729, 791, 812, 823, 904, 1010, 1124, 1207, 1210, 1851, 1938, 1943, 2196, 2292, 2304, 2676, 2820, 3086, 3097, 3230, 3318, 3453, 3753, 4607, 5610, 5625, 5984, 6081, 6560, 6702, 6756, 7251, 7676, 8675, 8703, 8999, 10230, 10866, 11646, 11664, 11903, 12884, 15218, 16806, 16849, 17328, 17384, 18649, 21588, 21609, 24965, 24987, 25765, 27630, 28182, 29196, 29737, 31212, 32882, 33857, 34199, 34566, 36840, 36864, 37513, 37887, 38134, 38239, 41545, 46212, 49409, 49461, 51293, 51762, 54101, 56503, 58462, 59022, 59049, 64165, 66167, 66198, 66465, 68010, 69479, 69709, 71852, 73627, 75263, 78244, 78529, 89970