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

Fermat Polygonal Number Theorem

In a recent post, I discussed centered tetrahedral numbers and so a recent Numberphile video naturally attracted my attention with the title 343867 and Tetrahedral Numbers. It turns out that there is a conjecture that every number can be written as the sum of at most five tetrahedral numbers and 343867 is the first number to require all five tetrahedral numbers. However, it can be represented by five tetrahedral numbers in 322 different ways. See Figure 1 where three of these ways are shown.

Figure 1: link

It should be borne in mind that the tetrahedral numbers are given by the formula:$$T_n=\frac{n \cdot (n+1) \cdot (n+2)}{6}$$The initial members are:

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180, 16215, 17296, 18424, 19600, 20825, 22100, 23426, 24804, 26235, 27720, 29260, 30856, 32509, 34220, 35990, 37820, 39711, 41664, 43680, 45760, 47905, 50116, 52394, 54740, 57155, 59640, 62196, 64824, 67525, 70300, 73150, 76076, 79079, 82160, 85320, 88560, 91881, 95284, 98770, 102340, 105995, 109736, 113564, 117480, 121485, 125580, 129766, 134044, 138415, 142880, 147440, 152096, 156849, 161700, 166650, 171700

Another interesting fact was mentioned in the video, namely that any number \(n\) can be represented by at most \(n\)-gonal numbers where \(n \gt= 3\). This means that any number can be represented by at most three 3-gonal (triangular) numbers, any number can be represented by at most four 4-gonal (square) numbers, any number can be represented by at most five 5-gonal (pentagonal) numbers and so forth.

This theorem is named the Fermat polygonal number theorem with the following details from Wikipedia:

The theorem is named after Pierre de Fermat, who stated it, in 1638, without proof, promising to write it in a separate work that never appeared. Joseph Louis Lagrange proved the square case in 1770, which states that every positive number can be represented as a sum of four squares, for example, 7 = 4 + 1 + 1 + 1. Gauss proved the triangular case in 1796, commemorating the occasion by writing in his diary the line "ΕΥΡΗΚΑ! num = Δ + Δ + Δ", and published a proof in his book Disquisitiones Arithmeticae. For this reason, Gauss's result is sometimes known as the Eureka theorem. The full polygonal number theorem was not resolved until it was finally proven by Cauchy in 1813.

Fermat Near Misses

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. This leads us to OEIS A050792:

A050792

Consider the Diophantine equation \(x^3 + y^3 = z^3 + 1  \, (1 < x < y < z) \) or 'Fermat near misses'. Arrange solutions by increasing values of \(z\) (see A050791). Sequence gives values of \(x\).

The initial \(x\) values are: 9, 64, 73, 135, 334, 244, 368, 1033, 1010, 577, 3097, 3753, 1126, 4083, 5856, 3987, 1945, 11161, 13294, 3088, 10876, 16617, 4609, 27238, 5700, 27784, 11767, 26914.

The corresponding \(z\) values are given by OEIS A050791:

A050791

Consider the Diophantine equation \(x^3 + y^3 = z^3 + 1 \, (1 < x < y < z)\) or 'Fermat near misses'. Sequence gives values of \(z\) in monotonic increasing order.

The initial \(z\) values are: 12, 103, 150, 249, 495, 738, 1544, 1852, 1988, 2316, 4184, 5262, 5640, 8657, 9791, 9953, 11682, 14258, 21279, 21630, 31615, 36620, 36888, 38599, 38823, 40362, 41485, 47584.

Given the \(x\) and \(z\) values, we can calculate the corresponding \(y\) values and so the initial \(x,y,z\) triples that satisfy \(x^3 + y^3 = z^3 + 1  \, (1 < x < y < z)\) are:

• (9, 10, 12)
• (64, 94, 103)
• (73, 144, 150)
• (135, 235, 249)
• (334, 438, 495)
• (244, 729, 738)
• (368, 1537, 1544)
• (1033, 1738, 1852)
• (1010, 1897, 1988)
• (577, 2304, 2316)
• (3097, 3518, 4184)
• (3753, 4528, 5262)
• (1126, 5625, 5640)
• (4083, 8343, 8657)
• (5856, 9036, 9791)
• (3987, 9735, 9953)
• (1945, 11664, 11682)
• (11161, 11468, 14258)
• (13294, 19386, 21279)
• (3088, 21609, 21630)
• (10876, 31180, 31615)
• (16617, 35442, 36620)
• (4609, 36864, 36888)
• (27238, 33412, 38599)
• (5700, 38782, 38823)
• (27784, 35385, 40362)
• (11767, 41167, 41485)
• (26914, 44521, 47584)

Specifically then we have \(26914^3+44521^3=47584^3+1\). The OEIS comments to A050792 are interesting:

Any number of solutions to the equation \(x^3 + y^3 = z^3 + 1\) can be produced through the use of the algebraic identity$$(9t^3+1)^3 + (9t^4)^3 = (9t^4+3t)^3 + 1$$by substituting in values of \(t\). Although these are certainly solutions, the identity generates only one family of solutions. Other solutions such as (64, 94, 103), (135, 235, 249) and (334, 438, 495) can be found. What is not known is if it is possible to parameterize all solutions for this equation. Put another way, are there an infinite number of families of solutions? Probable yes, but that too remains to be shown.

Here are the triples thrown up by the parameterized solutions for values of \(t\) from 1 to 20:

[9, 10, 12]
[73, 144, 150]
[244, 729, 738]
[577, 2304, 2316]
[1126, 5625, 5640]
[1945, 11664, 11682]
[3088, 21609, 21630]
[4609, 36864, 36888]
[6562, 59049, 59076]
[9001, 90000, 90030]
[11980, 131769, 131802]
[15553, 186624, 186660]
[19774, 257049, 257088]
[24697, 345744, 345786]
[30376, 455625, 455670]
[36865, 589824, 589872]
[44218, 751689, 751740]
[52489, 944784, 944838]
[61732, 1172889, 1172946] 
[72001, 1440000, 1440060] 

Folium of Descartes

René Descartes: 1596-1650

When I was searching for words ending in "-ium" for a Pedagogical Posturing blog that I came across the word "folium" and discovered its mathematical significance. To quote from Wikipedia:

In geometry, the folium of Descartes is an algebraic curve defined by the equation:$$x^{3}+y^{3}-3axy=0$$The name comes from the Latin word "folium" which means "leaf". The curve was first proposed and studied by René Descartes in 1638. Its claim to fame lies in an incident in the development of calculus. Descartes challenged Pierre de Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do. Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.

Pierre de Fermat: 1601-1665

Figure 1 depicts the shape of the graph when \(a=1\) as well as showing the line \(x+y+1=0\) which is asymptotic to it.

Figure 1: The folium of Descartes (green)
with asymptote (blue) when ${\displaystyle a=1}$

It is symmetrical about the line \(y=x\) and the two intersect at (0, 0) and (3\(a\)/2, 3\(a\)/2). The latter point is (1.5, 1,5) in Figure 1 where \(a\) has the value 1. When \(a=2\), the coordinates of point on the extremity of the loop takes on integer values viz. (3, 3). It's interesting to explore what values of \(a\) produce these points with integer values and how many there can be. Here is what I discovered:

• when \(a\) is an odd prime, there are no such points
• when \(a\) is composite, there is at least one such point
• when there is only one point, it is always the point at the extremity of the loop
• when there are an even number of points, none are at the extremity of the loop
• when there are an odd number of points, one of them is at the extremity of the loop

Here are the records for number of points:

• one point: \(a=2\) with point (3, 3)
  
• two points: \(a=3\) with points: 
• (2, 4), (4 ,2)
  
• three points: \(a=6\) with points: 
• (4, 8), (8,4), (9, 9)
  
• four points: \(a=63\) with points: 
• (42, 84), (80, 100), (84, 42), (100, 80)
  
• five points: \(a=42\) with points: 
• (5, 25), (25, 5), (28, 56), (56, 28), (63, 63)
  
• six points: none found thus far
  
• seven points: \(a=84\) with points: 
• (10, 50), (27, 81), (50, 10), (56, 112), (81, 27), (112, 56), (126, 126)
  
• eight points: none found thus far
  
• nine points: \(a=252\) with points: 
• (30, 150), (81, 243), (150, 30), (168, 336), (243, 81), (320, 400), (336, 168), (378, 378), (400, 320)

For values of \(a > 336\), the points occur only in pairs and increasingly sparsely. Figure 2 shows the situation for \(a=6\) where there are three pairs of integer coordinates.

Figure 2

Figure 3 shows the situation for \(a=42\) where there are five pairs of integer coordinates.

Figure 3

Some other interesting facts about the graph are:

• the area of the interior of the loop is found to be \(3a^{2}/2\), so in:
• Figure 1 the area is 1.5 square units
• Figure 2 the area is 54 square units
• Figure 3 the area is 2646 square units
  
  
• the area between the "wings" of the curve and its slanted asymptote is also \(3a^{2}/2\)
  
  
• Implicit differentiation gives the formula for the slope of the tangent line to this curve to be:$$ \frac{dy}{dx}=\frac{ay-x^2}{y^2-ax}$$
• the graph has a parametric form of:$$x=\frac{3ap}{1+p^3} \text{ and } y=\frac{3ap^2}{1+p^3}$$

Click the following links for biographies of Descartes and Fermat.

Generalised Fermat Primes

Today I turned 26188 days old and this number happens to be associated with the so-called generalised Fermat primes. Before going further, we should establish what is meant by a Fermat prime and a Fermat number. 

A Fermat number \(F_n\) is a number such that \(F_n=2^{2^n}+1\). 

If the number is prime, then we have a Fermat prime. Currently, only five such primes are known and these are:

 \(F_0=3, F_1=5, F_2=17, F_3=257, F_4=65537\)

A generalised Fermat number is a number of the form \(a^{2^n}+1\) where \(a>2\). Only if \(a\) is even can a generalised Fermat number be prime. Now 1024=\(2^{10}\) and so if we look at numbers of the form \(a^{2^{10}}+1\), we find that the values of \(a\) that produce primes are (up to 26188): 

1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188

These numbers form OEIS A057002. Figure 1 shows a plot of these same numbers:

Figure 1

Many of the largest known prime numbers are generalised Fermat numbers. To date (14th December 2020), the largest such prime is:$$1059094^{2^{20}}+1=1059094^{1048576}+1 \text{ which contains } 6317602 \text{ digits }$$This prime was discovered in 2018 but it pales in comparison to a number of larger Mersenne primes, the largest of which (discovered also in 2018) is:$$2^{82589933-1} \text{ which contains } 24862048 \text{ digits }$$A list of the current largest 100 primes can be found here.

It should be noted that there is another less common definition of a generalised Fermat number and that is:$$F_m(a,b)=a^{2m}+b^{2m} \text{ with gcd(\(a,b\))=1}$$I've looked at generalisations or extensions of other number types in the past, specifically:

• Generalised Cunningham Chains
• Fibonacci-like Sequences
• Beyond Fibonacci
• Root-Mean-Square and other Means

UPDATE on Thursday, February 4th 2021

Today I turned 26240 days old and one of the properties of this number, as with 26188 that is dealt with in this post, is that it is a generalised Fermat prime. Furthermore, both numbers belong to OEIS A057002:

A057002

Numbers n such that n^1024 + 1 is prime (a generalized Fermat prime).

In fact, looking at the members of the sequence, we see that there is a cluster of three numbers (26132, 26188, 26240) and Figure 2 makes this even more apparent:

1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188, 26240, 29074, 29658, 30778, 31126, 32244, 33044, 34016, ...

Figure 2: cluster of generalised Fermat primes

The abc Conjecture and Fermat's Last Theorem

An article appeared in Scientific American's website on April 3rd 2020 titled Mathematical Proof That Rocked Number Theory Will Be Published and which began:

Figure 1: Shinichi Mochizuki

After an eight-year struggle, embattled Japanese mathematician Shinichi Mochizuki has finally received some validation. His 600-page proof of the abc conjecture, one of the biggest open problems in number theory, has been accepted for publication. 

Acceptance of the work in Publications of the Research Institute for Mathematical Sciences (RIMS)—a journal of which Mochizuki is chief editor, published by the institute where he works at Kyoto University—is the latest development in a long and acrimonious controversy over the mathematicians’ proof. 

Two other RIMS mathematicians, Masaki Kashiwara and Akio Tamagawa, announced in Japanese the publication at a 3 April press conference in Kyoto. The paper “will have a big impact”, said Kashiwara. When asked how Mochizuki reacted to news of the paper’s acceptance, Kashiwara said, “I think he was relieved.” 

Mochizuki, who has denied requests for interviews over the years, did not appear and did not make himself available to reporters. 

Eight years ago, Mochizuki posted four massive papers online, claiming to have solved the abc conjecture. The work baffled mathematicians, who spent years trying to understand it. Then, in 2018, two highly respected mathematicians said they were confident that they had found a flaw in Mochizuki’s proof—something many saw as death blow to his claims. 

The latest announcement seems unlikely to move many researchers over to Mochizuki’s camp. “I think it is safe to say that there has not been much change in the community opinion since 2018,” says Kiran Kedlaya, a number theorist at the University of California, San Diego, who was among the experts who put considerable effort over several years trying to verify the proof. Another mathematician, Edward Frenkel of the University of California, Berkeley, says, “I will withhold my judgment on the publication of this work until it actually happens, as new information might emerge.”

The article goes on to describe the abc conjecture in these terms:

The ‘abc conjecture’, the problem Mochizuki claims to have solved, expresses a profound link between the addition and multiplication of integer numbers. Any integer can be factored into prime numbers, its ‘divisors’: for example, 60 = 5 x 3 x 2 x 2. The conjecture roughly states that if a lot of small primes divide two numbers \(a\) and \(b\), then only a few, large ones divide their sum, \(c\). 

Let's look at a different definition now, taken from Wikipedia:

Take three positive integers, \(a\), \(b\) and \(c\) (hence the name) that are relatively prime and satisfy \(a + b = c\). If \(d\) denotes the product of the distinct prime factors of \(a \times b \times c\), the conjecture essentially states that \(d\) is usually not much smaller than \(c\). In other words: if \(a\) and \(b\) are composed from large powers of primes, then \(c\) is usually not divisible by large powers of primes.

What's highlighted in red is essentially saying the same thing and provides an approach to proving Fermat's Last Theorem. To see this, let's look at \(a=13^{22}\) and \(b=11^{22}\) as an example. We know that Fermat's Last Theorem hopes to find a value for \(n>2\) such that:$$a^n+b^n=c^n \text{ for integer values of }a,b,c$$However, the \(abc\) conjecture indicates that for \(13^{22}+11^{22}\) this is highly unlikely. In fact, we have:$$13^{22}+11^{22}=2 \times 5 \times 29 \times 44617 \times 955769 \times 266300690657$$Here is another example:$$101^{19}+197^{19}=2 \times 149 \times 229 \times 1483 \times 110398608667213673 \times 3521300862956110103$$So from those two examples, one gets a glimpse of how a generalised approach using the \(abc\) conjecture might be used to provide a proof of Fermat's Last Theorem.

Of course, neither of the previous two definitions is a formal definition of the conjecture. We'll get to that shortly. In the meantime, here is a Numberphile video that appeared soon after Mochizuki's paper appeared:

In the video, the conjecture is described as follows where rad is the product of the distinct prime factors:

If \(a\) and \(b\) are coprime integers and \(a\)+\(b\)=\(c\) then (in general):$$ \text{rad}(abc)^k>c$$If \(k=1\) there are infinitely many exceptions.

If \(k>1\) there are finitely many exceptions.

He quotes \(3 + 125 =128\) as an example of an exception when \(k=1\). In this case, we have \( \text{rad}(3, 125, 128) = 3 \times 5 \times 2 = 30 < 128\). For the examples that I used earlier however, the opposite is the case. Figure 1 shows a SageMathCell screenshot for \(13^{22}+11^{22}\), with permalink included:

Figure 1: permalink

The definitions can get more technical than those quoted earlier but that's probably enough for now.