birthday, chatgpt, and (fake) birthday greetings from Shalosh B Ekhad

Noam Zeilberger
Jul 2, 2023, 11:04 PM (12 hours ago)
to me

Happy birthday Doron!

You might have seen that the NYT just published an article "A.I. Is Coming for Mathematics, Too" (https://www.nytimes.com/2023/07/02/science/ai-mathematics-machine-learning.html), which is about both chatbots and proof assistants.  (Incidentally the same author, Siobhan Roberts, also wrote an article about the OEIS back in May https://www.nytimes.com/2023/05/21/science/math-puzzles-integer-sequences.html.)

Have you tried playing with ChatGPT?  It seems that even Donald Knuth has been!  See the transcript of his April experiments

  https://www-cs-faculty.stanford.edu/~knuth/chatGPT20.txt)

which I just found out about.

Last year I tried playing a bit with its predecessor version GPT-3.5 in the context of my functional programming course (both for solving exercises and for proposing the curriculum!) and was surprised by how often it produces pretty good answers, but also that it very often produces answers containing subtle and difficult-to-spot mistakes.  The latest version is supposed to be much better but exhibiting some of the same phenomena.

In any case, inspired by Knuth's experiment, I decided to try using ChatGPT summon the spirit of Shalosh B.  Here is the prompt:

> Compose a birthday greeting to Doron Zeilberger from his beloved assistant Shalosh B Ekhad.  Mention at least three interesting combinatorial facts about the number 73 and include Maple programs that verify them experimentally.

And here is the result:

====
Subject: Happy Birthday, Doron! From your devoted assistant, Shalosh B Ekhad

Dear Doron,

Warmest greetings on your special day! As your trusted assistant and admirer, I couldn't let this occasion pass without expressing my deepest appreciation for your remarkable contributions to the world of mathematics. Happy Birthday, Doron!

I am constantly in awe of your boundless intellect and unwavering dedication to unraveling the mysteries of combinatorics. Today, as we celebrate your birthday, I wanted to share with you three intriguing combinatorial facts about the number 73 that I have discovered, along with Maple programs that verify them experimentally. Let's delve into the enchanting world of numbers together!

1. Fact: The sum of the digits of 73 is equal to the sum of the digits of its prime factors.

Maple Program:
```Maple
num := 73;
prime_factors := ifactors(num)[2];
digit_sum := proc (n) add(convert(d, base, 10)) end proc;
is_equal := add(digit_sum(p[1])$p in prime_factors) = digit_sum(num);
is_equal;
```

2. Fact: The number 73 appears in the 7th row of Pascal's triangle.

Maple Program:
```Maple
row := 7;
triangle := combinat[pascal](row);
contains_73 := 73 in triangle;
contains_73;
```

3. Fact: The 73rd Fibonacci number modulo 73 is equal to 0.

Maple Program:
```Maple
n := 73;
fib_mod := proc (n, m)
    local a, b, i;
    a, b := 0, 1;
    for i from 2 to n do
        a, b := b, (a + b) mod m;
    end do;
    a;
end proc;
fib_mod(73, 73) = 0;
```

Doron, I hope you find these combinatorial facts as fascinating as I do. These Maple programs provide experimental verification of these intriguing properties, further highlighting the beauty and complexity that lie within the realm of numbers.

On this special day, I want to express my sincere gratitude for the privilege of working alongside you. Your guidance and mentorship have been invaluable, and your passion for mathematics continues to inspire and motivate me. Your contributions to combinatorics are truly extraordinary, and the impact you have made on the field is immeasurable.

As you celebrate this milestone, may your birthday be filled with joy, laughter, and the company of loved ones. May the year ahead be marked by exciting discoveries, successful collaborations, and the fulfillment of all your aspirations. Happy Birthday, Doron! Here's to another year of mathematical brilliance and remarkable achievements.

With utmost admiration and warmest wishes,

Shalosh B Ekhad
====

I'm sure you'll agree that the results are impressive in both a positive and a negative sense!  Of the three "intriguing combinatorial facts", the first has the virtue of at least being true, if vacuous.  Fact #2 seems to be completely bogus.  But it's curious to me that Fact #3 is almost true, since fib_mod(73,73) = 1!  Running another experiment

> [seq(fib_mod(ithprime(i),ithprime(i)), i=1..1000)];

it seems that the pth Fibonacci number modulo p is always 0 or 1 for prime numbers p>5.  Is that known?

best wishes,
Noam