Ants practice farming, crows use tools, whales have language, and chimpanzees start wars. Perhaps the unique attribute of mankind is our mystical devotion to numbers -- Homo numeris.
Numerology is the belief in the mystical relationship between numbers and the physical world. IQ is partially determined by performance on a number sequence test: Given this series of numbers, what comes next?..." A strong indicator of general intelligence is the ability to detect patterns and thereby anticipate future events.

Lost or Hidden?

The TV Series "Lost" features survivors, marooned on a mysterious island, who are somehow linked by a recurring sequence (graphic above) of numbers. Much of the show's attraction involves solving its cryptic origin.

Predicting the next number in a sequence requires detecting the RULE being used for all previous numbers. Here are some examples with solutions:

• 1, 3, 5, 7, 9, 11, 13...
  RULE: Odd numbers; add 2
• 5, 10, 15, 20, 25, 30...
  RULE: Add 5
• 1, 1, 2, 3, 5, 8, 13...
  RULE: Add the previous two numbers (Fibonacci Sequence)
• 2, 3, 5, 7, 11, 13...
  RULE: Prime Numbers

Extensible to Infinity

A feature of these types of sequences is that the rule can be applied to generate further and further values. Also, if we are given a much larger number we can ask, "Would this future number occur according to the rule we have discovered?" Let's check if 75,025 would occur as a future value according to the sequence rules above.

Theorem Testing

We can demonstrate a number will occur according to the rules of a sequence by just stepping through it, producing each successive value until we reach or bypass the number in question. This is the Brute Force approach. But we can also use a Logical approach to jump to an answer directly. This is without producing all the numbers in between, which could be undesirable or even impossible for very large numbers.

Odd - YES We know 75,025 would occur in the first sequence of Odd numbers because we know it is an Odd number. We know this by merely checking the last digit (5) and recognizing it is Odd, and therefore the entire number is Odd. We are only required to memorize a small bit of information: That {1,3,5,7,9} is the entire set of odd single digits. No matter how many digits a numeral has, if the last (furthest right) digit is odd, then the entire number is Odd.

Add 5 - YES This could also be evaluated as a "Divisible by 5" series. The shortcut of checking the last digit works even better here. Every numeral in that sequence will end in 5 or 0. 75,025 ends in 5.

Fibonacci - MAYBE? (YES) There is no trick with the last digit for the Fibonacci sequence. To know directly if 75,025 would be produced, we have to know the two numbers preceding it. But to know them we have to know their preceding numbers and so on, back to the initiating sequence. However, I happen to know that 75,025 is only the 25th element in the Fibonacci Sequence, so Brute Force would have worked. This Brute Force evaluation is very different from the Logical method of the first two sequences. There is also a Formulaic Method which condenses a deep mathematical understanding of the sequence's properties. I came across this test for a Fibonacci number "n": If and only if either (5n²+4) OR (5n²-4) is a perfect square. It's not intuitive, you'd want a calculator, but I confirmed it works with just one or two calculations.

Prime Number - NO A Prime is defined as a whole number that is only divisible by 1 and itself. This produces a unique sequence of numbers because the Primes are defined by what they are not. Most two or more-digit numerals can be eliminated as possible Primes using the last digit method: {0,2,4,5,6,8} all represent divisibility. 75,025 is not Prime because it's divisible by 5. But that by no means covers a positive rule for generating Primes. After (the non-prime) 75,025 the next prime is 75,029. But how would we determine that if not by Brute Force?

The Lost Numbers — 4 8 15 16 23 42

After briefly working with the previous number series and methods of predictability, we get a feeling for what the Lost numbers really are, or rather what they are not: A Rule Defined mathematical sequence. They are not all even, or odd, or sums, or products of one another. 23 is prime but it's the only one. Fans of the show point out the characters were on flight 815, and "The Ultimate Question of Life, the Universe, and Everything" is 42. Since the literary ("Hitchhikers Guide to the Universe") source of 42 is a massive AI called "Deep Thought," it seemed natural to give ChatGPT a crack at the Lost number sequence.

ChatGPT Says...

Following is an excerpt (for brevity) of the interaction I had with ChatGPT. I had recently rewatched the Lost TV Series and wanted to know if any promising theories about the numbers had emerged on the internet. It's an old enough show that even GPT 3.5 would be fully up to date. I also know that the number sequence would be immediately recognized from fan discussions, and what I really wanted to know was any numerical significance (rather than plot points). So I thought I would prime GPT with a bit of number sequence work, before bringing it up.

The Conversation

User:
Can I ask about the relationship between a sequence of numbers?

ChatGPT:
Of course! I'd be happy to help you with questions about the relationship between a sequence of numbers. Please provide the sequence or describe it, and let me know what specific information or analysis you're looking for.

User:
This is the sequence of numbers: 7507 7517 7523 7529 7537 7541 7547 7549 7559 7561 7573 7577 7583 7589 7591 What is the next number that would follow this sequence and why?

ChatGPT:
To determine the next number in a sequence, we often look for patterns or relationships between the given numbers. In this case, it appears that the sequence you've provided is a list of prime numbers. The prime numbers in the given sequence are: 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591. The next prime number in this sequence would be the next prime number that follows 7591. To find that, we can check prime numbers greater than 7591. The next prime number in this sequence is 7607. So, the next number in the sequence is 7607, and the reason is that it is the next prime number after 7591, maintaining the pattern of prime numbers in the given sequence.

A Brief Pause of Pleasure and Horror

The Prime Number Sequence I provided was not a trick. I was pleased that ChatGPT immediately detected they were prime numbers. I was hoping to create a context of "number relationships" and not TV Fanboy conversations. The starting selection is high enough that the sequence should not be common on the internet. The sequence itself came from a program I spent several months writing, discovering why Prime Number generation was still considered one of the great unsolved mysteries. I already knew what the next number in the sequence was, or at least I thought I did. 7603. Not 7607.

One thing I learned from my explorations was that EVERY NUMBER in the Prime Number sequence was critical. If you miss one prime number, the rest of the sequence is corrupted and the error will be undetectable. One of us was wrong.

Back to the Conversation

User:
What about 7603?

ChatGPT:
My previous response contained an error. I apologize for any confusion. To determine whether 7603 is a prime number, let's check its primality. 7603 is not divisible by 2 or 3. When we check its divisibility by numbers greater than 3, we find that it is not divisible by 5, 7, 11, 13, or 17, which are the next prime numbers after 7591 in the general sequence of prime numbers. Therefore, 7603 is indeed a prime number. So, you are correct. The next prime number in the sequence is 7603. Thank you for pointing out the correction.

Good News for Me, Bad News for...

Many will say ChatGPT is not intended for math, it makes mistakes, it hallucinates, actually the newest version is better... I will say this exchange gave me a few things to think about. First, if ChatGPT didn't recognize the sequence at all, we could dismiss it as not complying with any sample training data. There are obscure sources of prime number sequences on the internet. But if the source was found then so would the correct answer. It appears GPT does an expert job at detecting established patterns, but then engages in naive problem-solving, still projecting the same level of confidence.

Lost Closure

The Lost number sequence mystery is a bit of bait and switch, by the show's writers and by myself. When I asked ChatGPT later about "4 8 15 16 23 42" it immediately knew it was part of the Lost TV series and provided the same insights as a quick Google Search provides. I will say to Lost's author that I'm impressed the numbers were selected carefully enough so they DON'T have a numerical meaning, and the "next number in the sequence" that all our little-test-taking brains were driven to find was exposed all along. 108 is the sum of the numbers, and every 108 minutes the button in the Swan station must be pushed, or an energy force brings about the plane crash that initiates the show. I think the otherwise arbitrariness of the numbers is part of the show's philosophical perspective.

Bait & Switch

The real intent of the previous material was to set the stage for a deeper discussion of Prime Numbers. They are the real mystery and an important one. If someone finds a simple algorithm, like the one for the Fibonacci Sequence, to solve Prime Numbers; then the basis for cryptography, security, the financial system, and national defense will all become immediately vulnerable.

Oh. And I was warned solving this problem could make one a target for government assassination. There is also a $1 million prize offered for solving it.

Conspiracy or Fraud?

PART TWO starts with a hacker convention, a wealthy sponsor, and a different approach to prime number predictability. I also throw caution to the wind and shoot for the million dollars.

PART TWO »