I try to avoid reading the AI Overviews when I do Google searches that show them. But this time I happened to glance at the AI Overview. And it was blatantly false.
I searched for [sum of three “consecutive cubes”]. Here’s what the AI Overview told me:
The sum of three consecutive cubes is always equal to the square of the sum of those three numbers; in other words, if you take any three consecutive integers, add them together, then square that sum, you will get the same result as adding the cubes of those three numbers.
Example:
• Consider the numbers 1, 2, and 3.
• Adding them gives: 1 + 2 + 3 = 6
• Squaring the sum gives: 6^2 = 36
• Adding their cubes gives: 1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36Key points:
• This property holds true for any set of three consecutive integers.
• You can use this fact to quickly calculate the sum of three consecutive cubes without having to individually cube each number.
It’s true that this pattern holds true if you happen to pick the numbers 1, 2, and 3. Super cool!
So let’s look at the very next set of three consecutive integers: 2, 3, and 4.
2 + 3 + 4 = 9, and 9^2 = 81.
2^3 + 3^3 + 4^3 = (8 + 27 + 64) = 99.
But wait, 81 doesn’t equal 99! What’s wrong???
What’s wrong is that the AI Overview’s claim is false for all cases except for (1,2,3), (0,1,2), and (-1,0,1). There’s one other case where the sum of the cubes of three consecutive integers is a perfect square (of a different number), but the only cases where it’s specifically the square of the sum of those three consecutive integers are (1,2,3), (0,1,2), and (-1,0,1).
(I’m reminded of a joke proof that all odd numbers are prime: 3 is prime; 5 is prime; 7 is prime; therefore, by induction, all odd numbers are prime.)
Google’s AI Overview links to sources to support its statements. The sources that it linked to in this case are the following. Not one of them makes the claim that the AI Overview is making.
- https://www.chegg.com/homework-help/questions-and-answers/q33-use-mathematical-induction-prove-sum-cubes-three-consecutive-natural-numbers-multiple–q107719986
- https://iitutor.com/proving-sum-of-consecutive-cubes-formula/ (“The sum of [the first] n consecutive cube numbers [starting with 1] is equal to the square of the [first] n numbers [also starting with 1].” That’s nifty, but it’s a very different claim than Google’s.)
- https://www.youtube.com/watch?v=csAL6zQ60mc
- https://proofwiki.org/wiki/Sum_of_Cubes_of_3_Consecutive_Integers_which_is_Square
- https://homework.study.com/explanation/prove-that-the-sum-of-cubes-of-any-three-consecutive-natural-numbers-is-divisible-by-9.html
As usual, the moral of this story is: Don’t believe anything that generative AI tells you.
I often remember to add ” -ai” to the ends of searches (to tell Google not to give an AI Overview), but I often don’t.
But I’ve been hearing reports that that isn’t working any more for some people. And, indeed, when I re-run this search with -ai, sometimes it gives me an AI Overview and sometimes it doesn’t.
Interestingly, the specific contents of the AI Overview vary—if I do the search without -ai, I get the one that I posted about, but if I do the search with -ai, I get a different claim (that doesn’t mention squares) that I haven’t checked yet.
Adding swear words does still seem to work to remove the AI Overview. It also provides search results that use those swear words, which may or may not improve your search results, depending on what kinds of search results you want.
The reason I was doing this search was that an interesting fact about a particular number was mentioned in a recent TV show. I had played around with the relevant numbers a bit, and had reached the point where I was curious about what work had been done on this topic.
If I hadn’t tried out the sums of cubes of three consecutive numbers on my own just prior, I might have been tempted to believe what the AI Overview said. But because I had just calculated several such answers myself, I knew immediately that the AI Overview was wrong.
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