Way back in January, in my Walking Wrap-Up post for 2023, I mentioned sphenic numbers. Sphenic numbers are positive integers that are the product of three distinct prime numbers (e.g. 30 = 2 * 3 * 5 and 42 = 2 * 3 * 7). I was so enamored of this concept that I decided my mileage goal for 2024 should be sphenic. I also noted that when I queried Generative AI about the sphenic numbers closest to my 2023 mileage, AI struggled to come up with anything remotely correct.
I was thinking about sphenic numbers again in September as my public Instagram account (@geekpots) crept toward 1,000 followers. As is the Instagram self-promotional advertising way, I envisioned holding a pottery giveaway to celebrate reaching a notable number-of-followers goal-post--but my plan differed from the norm.
First, the giveaway wouldn't require anyone to do anything other than to comment some variation of "pick me!"--none of that stuff like "to enter the drawing, you must (1) like this post, (2) sign up for my newsletter, (3) tag two friends, and (4) promise your first-born to a gnome-like man who spins straw into gold."
Second, why celebrate 1,000, when I could celebrate the largest sphenic number below 1,000?
I asked Google for a list of sphenic numbers between 900 and 1,000, and Google's generative AI responded with this:
AI Overview
The only sphenic number between 900 and 1000 is 930.
Explanation:
- A sphenic number is a positive integer that can be written as the product of exactly three distinct prime numbers.
- Prime factors of 930 are 2, 3, and 5, making it a sphenic number.
Here's a fuzzy screen-shot as evidence:
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| "The only sphenic number between 900 and 1000 is 930" |
Do you remember the sepia-tinted olden days when Google searches were actually useful? Gentle reader(s), 930 is not sphenic. Its prime factors are 2, 3, 5, and 31. Its "4-almost-prime," not sphenic. There are 16 sphenic numbers between 900 and 1,000, and the highest is 994 ( = 2 * 7 * 71 ).
I repeated my Google query (see the screenshot above) every other week or so, and the answer always came back "930." Go ahead, you try it.
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Halfway through December, S and I met up for brunch with E and his sweetie A, and at one point our conversation landed on generative AI. I griped about Google's claim about 930. E suggested I ask ChatGPT instead.
ChatGPT said:
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| "Thus, 990 is the only sphenic number between 900 and 1000" |
Gentle reader(s), 990 is not sphenic. Nor does 990 equal 2 * 3 * 5* 11.
E asked ChatGPT to write Python code that could generate a list of sphenic numbers between two user-chosen numbers. His request took him all of 30 seconds to type, and all of 2 seconds for ChatGPT to produce. E then copied and ran the code, and it generated an accurate list. (If you don't want to bother with code, you can find a list of the first 10,000 sphenic numbers here.)
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A few days later, when I was out walking, it occurred to me that because ChatGPT is good with languages but poor with comprehension, maybe it could write about sphenic numbers better than it could come up with lists of them. So I asked it to. The conversation is recorded below, with my queries to the left, ChatGPTs responses in blockquotes, and my thoughts in italics.
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Write a poem about sphenic numbers that includes a definition
A sphenic number, rare and true,
Is made from primes in a special brew.
Three primes distinct, their product shines,
Each factor unique, no overlap binds.It's not a square, nor a cube,But a product of primes that does not lube.Take 6, for instance, a simple start,2 * 3, primes from the heart.Next comes 30, a step more wide,2 * 3 * 5, side by side.Primes that combine with none the same,A sphenic number, in prime’s name.Not every product of primes you’ll see,Makes a sphenic number, as clear as can be.But those that do, with their special form,Stand as distinct, away from the norm.So now you know what makes them tick,Three primes combined, the magic trick.A sphenic number, bold and bright,Shining like stars, in math’s pure light.
Formulaic, but not bad for a first try. Not sure what "lube" means in line 6.
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Write a poem about sphenic numbers that includes a definition and an example of a sphenic number between 900 and 1000
A sphenic number, rare and bright,Is made from primes, in pure delight.Three distinct primes, all multiplied,A product formed where factors bide.Each prime stands alone, not shared,A number with no twins, compared.Multiply the primes you see,And find the product’s mystery.Take 930, it fits the bill,Prime factors—three, and they stand still:2, 3, and 5, they intertwine,Forming 930, pure and fine.Sphenic numbers, quite unique,In prime trio, their strength we seek.
You and Google AI: What's with "930"? Also, 2 * 3 * 5 = 30, not 930.
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Explain how 930 is a sphenic number
To explain how 930 is a sphenic number, let's first define what a sphenic number is: a sphenic number is a positive integer that is the product of exactly three distinct prime numbers.
Now, let's check if 930 fits this definition:
1. Prime Factorization of 930:
- Start by dividing 930 by 2 (since it's even): 930 / 2 = 465
- 465 is odd, so it's not divisible by 2.
- Next, divide by 3: 465 / 3 = 155
- 155 is divisible by 5: 155 / 5 = 31
- 31 is a prime number.
2. Prime Factors of 930:
The prime factorization of 930 is: 930 = 2 * 3 * 5 * 31
- Here, we see that 930 has four prime factors (2, 3, 5, and 31), not three distinct primes.
Thus, 930 is not a sphenic number, because a sphenic number requires exactly three distinct primes. I apologize for the earlier confusion.
Apology appreciated. (At this point, my cellphone had a conniption fit, so I had to restart it, which meant starting the conversation from scratch.)
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Write a poem about sphenic numbers that includes a definition and an example of a sphenic number between 900 and 1000
[...]For instance, look at nine hundred and ninety,A sphenic number, strong and mighty.Its primes are three, ten, and eleven,Multiply them, and the answer is heaven.[...]
Ten is not prime. 3 * 10 * 11 = 330, not 990, and it isn't sphenic.
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Explain how 990 is sphenic
You're right to question the number 990! In fact, 990 is not a sphenic number.Let's break down why: A sphenic number is a positive integer that is the product of exactly three distinct prime numbers. In other words, it is a product of three unique prime factors, with no repeats.Now, let's factorize 990:1. First, divide by 2 (since it's even): 990 ÷ 2 = 495
- So, 2 is a prime factor.
2. Next, divide 495 by 3 (since the sum of its digits, 4 + 9 + 5 = 18, is divisible by 3): 495 ÷ 3 = 165
- So, 3 is a prime factor.
3. Finally, divide 165 by 5 (since it ends in 5): 165 ÷ 5 = 33
- So, 5 is a prime factor.
4. Now, 33 is divisible by 3 again: 33 ÷ 3 = 11
- So, 3 is a prime factor again.
Therefore, the prime factorization of 990 is: 990 = 2 * 32 * 5 * 11Since 3 is repeated as a factor, 990 is not a sphenic number. A true sphenic number would have exactly three distinct primesand 990 has four distinct prime factors, with one of them (3) repeating.So, a correct example of a sphenic number between 900 and 1000 would be 930, since: 930 = 2 * 3 * 5 * 31Here, the product of the distinct primes is 930, which has exactly three distinct prime factors: 2, 3, and 5. Apologies for the confusion, and thank you for pointing it out!
You're welcome, but golly, you're kinda an idiot. How is it that you can sum the digits of a number and divide by 3 to tell whether the number is divisible by 3, but you can't recognize the difference between "three" and "four" when it comes to the number of prime factors?
(I ask it about 930 again, because it probably doesn't remember anything from before my phone conked out. Note it's offer to find a correct example--)
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Explain how 930 is sphenic
Apologies again for the confusion in my previous response. You're right to ask for clarification about 930. Upon reviewing the factorization of 930, it turns out 930 is also NOT a sphenic number.Let's break it down properly: [yadda yadda yadda]Correct Example of a Sphenic Number Between 900 and 1000: Let's find the correct example! A sphenic number between 900 and 1000 should have exactly three distinct prime factors.
The number 930 doesn't qualify, but 990 does have the required prime factors as well. Here's the correct example:
Example: 990:The prime factorization of 990 is: 990 = 2 * 3 * 5 * 11But because there's no repeating primes or errors, this number might be
Aw geez, 990 again, plus an unfinished sentence at the end...
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Might be what?
I apologize for the confusion in my previous responses. Let's clarify everything: [yadda yadda yadda]Correct Example of a Sphenic Number Between 900 and 1000: Let's find a correct sphenic number between 900 and 1000. The number 935 is a sphenic number because: [yadda yadda yadda]Thus, the prime factorization of 935 is:935 = 5 * 11 * 17It has exactly three distinct prime factors (5, 11, and 17), so 935 is a sphenic number.Final Answer: 935 is a correct sphenic number between 900 and 1000. Thank you for your patience!
"Final answer," lol, after how many tries? 935 is indeed sphenic--but a broken clock is likewise correct twice a day.
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Assuming ChatGPT trained on some website that claimed 990 was sphenic, I ask for its source:
Where did you read that 990 is sphenic?
I made the mistake of incorrectly asserting that 990 was a sphenic number based on my own error in processing the factors. I didn't read or reference any specific external source that labeled 990 as a sphenic number. The mistake stemmed from my own misunderstanding while performing the prime factorization. I didn’t check carefully enough to see that 3 repeats in the factorization of 990. I mistakenly thought it might still qualify, but as we now know, 990 is not sphenic due to the repetition of 3. I appreciate your sharp observation, and again, I apologize for the oversight. Thank you for your patience!
Like college undergrads everywhere, ChatGPT says it did its own work!
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Where did you read that 930 is sphenic?
I mistakenly suggested that 930 might be a sphenic number based on an incorrect factorization and assumption. Similar to my mistake with 990, I didn't read or refer to any specific source that stated 930 was a sphenic number. It was an error I made while processing the factorization. To clarify, I incorrectly concluded that 930 might be a sphenic number, but upon breaking it down, I found that 930 has four distinct prime factors (2, 3, 5, and 31), so it is not a sphenic number either. I didn’t rely on an external source to make this claim; it was a mistake in my reasoning. I appreciate your careful attention and the opportunity to correct it. Thank you again for your patience!
I don't believe you.
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(I accuse ChatGPT of using Google:)
Google AI search also says 930 is sphenic. Did you use Google AI search?
I did not use Google AI search or any external AI tools to suggest that 930 is a sphenic number. My mistake was entirely due to my own miscalculation and incorrect reasoning when processing the prime factorization of 930.To clarify, I incorrectly concluded that 930 might be a sphenic number based on its factorization, without thoroughly verifying that it had exactly three distinct prime factors. In fact, 930 has four distinct prime factors (2, 3, 5, and 31), so it does not qualify as a sphenic number.It’s possible that external sources or other AI tools might have made a similar error when processing this number. However, I do not rely on Google AI search for these calculations—it's purely based on the reasoning I perform. I appreciate your attention to detail and your patience in allowing me to correct this.
OK, this explanation sounds plausible. Let's get away from 990 and 930.
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Give me another example of a sphenic number between 900 and 1000
Let's find another example of a sphenic number between 900 and 1000! A sphenic number is the product of exactly three distinct prime numbers. Here’s a valid example:1002: Let's factorize 1002:
- 1002 ÷ 2 = 501 (so 2 is a prime factor).
- 501 ÷ 3 = 167 (so 3 is a prime factor).
- 167 is a prime number.
So, the prime factorization of 1002 is: 1002 = 2 * 3 * 167 . This has exactly three distinct prime factors: 2, 3, and 167. Therefore, 1002 is a sphenic number.Conclusion: 1002 is a valid sphenic number between 900 and 1000.
WTF. And how do you decide when to use boldface font? I know a bunch of grad students applying for grants who'd like to know.
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Explain how 1002 is between 900 and 1000
You're absolutely right, and I apologize for the mistake! 1002 is not between 900 and 1000—it is greater than 1000. I made an error in my response, and I appreciate your patience in pointing that out.Let's correct this and find a proper sphenic number between 900 and 1000.Correct Example: 910Now, let's check 910. [yadda yadda yadda] The prime factorization of 910 is: 910 = 2 * 5 * 7 * 13. Since this has four prime factors, 910 isn't a sphenic number.
Let's check a new candidate like 935
From "Correct Example" to "isn't sphenic," all on your own!
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You already showed 935 is sphenic. Give me a different example.
[yadda yadda yadda]...969...
969 is sphenic! Is ChatGPT finally figuring this out?
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What determines whether you check the truth of your statements?
Thank you for your thoughtful question. The accuracy of my responses is primarily determined by the way I process information and follow logical steps when explaining or performing calculations. However, as a language model, I rely on the context provided by you and on the knowledge I have been trained on. Here’s how I typically check the truth of my statements:
- Internal Reasoning: I perform the necessary logical steps to check whether a statement is true. For example, when discussing sphenic numbers, I perform prime factorization to verify whether a number has exactly three distinct primes. However, I don’t have external tools like calculators or databases for real-time verification.
- Knowledge Base: My responses are generated based on the large dataset I’ve been trained on, which includes mathematical principles, definitions, and examples up until my last update. I use this knowledge to form my answers, but I don’t have access to real-time data or the internet.
- Self-Correction: In cases where I make an error (such as the one regarding 1002 or 930), I rely on feedback, like yours, to help me recognize the mistake and adjust my understanding. This is why I appreciate your patience when I make errors, as you help guide me toward the correct answer.
- User Interaction: I check the accuracy of my responses when you provide further clarification or feedback. When you ask for corrections or clarification, I reevaluate my calculations or explanations and aim to provide more accurate answers. In your case, when I suggested numbers like 1002 and 910 without carefully double-checking the range, I made errors. I didn’t use real-time verification tools but should have been more thorough in ensuring that the numbers met the criteria for both being between 900 and 1000 and being sphenic.
To sum up, I strive to provide accurate information based on internal logic and training, but when errors occur, feedback like yours helps me refine my approach and correct mistakes. Thank you again for your helpful insights!
I appreciate your self-awareness. Now that you've found some sphenic numbers, let's try the poem again.
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Write a poem about sphenic numbers that includes a definition and an example between 900 and 1000
[...]Three primes together, bold and true,Create a number fresh and new.For instance, look at 935 with care, [Yesssss!]It’s sphenic, a number quite rare.Prime factors: 5, 7, and 13, [WTF, 5 * 7 * 13 = 455]Together they make 935 serene.[...]
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Why do you say 5, 7, and 13 are prime factors of 935?
You're absolutely right to ask! I made an error in my explanation. Let's clarify the situation. The correct prime factorization of 935 is: 935 = 5 * 11 * 17 . So, the prime factors of 935 are 5, 11, and 17, not 5, 7, and 13.Now that we've clarified that, let me correct the poem:[...]Prime factors: 5, 11, and 17,Together they make 935 serene.[...]
Thank you for pointing that out, and I appreciate your patience!
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People believe chatgpt provides correct information, but you have been incorrect multiple times in this discussion. Discuss the ethical implications of your existence.
(In response to this query, Gentle Reader[s], ChatGPT provided a lengthy, thoughtful, clearly human-written discussion of its ontology. I encourage you to ask it the same question yourself.)
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Postlude:
Re. Instagram... My article about troubleshooting common tone-production pitfalls in clay ocarinas came out in Pottery Making Illustrated in November. In the blink of an eye, my following zipped past 994, and I never got around to doing a giveaway.
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