Written: April 1 (!), 2013
It is ironical that mathematicians, who claim to be so enamored with absolute truth- hence their obsession with the quaint notion of rigorous proof- are pretty good at hiding the truth from their peers, and the general educated public, when they think (erroneously!) that the revealed facts would either tarnish the image of their demi-gods (like Gauss), or cast doubts on their rigid dogmas.
I am talking about the very plausible rumors that a noted French mathematician (whose name I am withholding for obvious reasons, so let's call him monsieur le professeur X, or Pr. X, for short), found, in his attic, a diary that must have been written by the great Gauss. According to this diary, Gauss had a love affair with Sophie Germain, and together they had a "natural daughter" (fille naturelle), that was put up for adoption. Pr. X, who recently discovered the diary in his attic, is apparently the great-grandson of that daughter-born-out-of-wedlock, and as a corollary, the great-great-grandson of two of the greatest mathematical giants of 19th-century math, (of both sexes!)
But the mores have changed since the times of Carl Friedrich and Sophie, and by today's standards, it is hardly a "scandal" to have a child out of wedlock, even if you are mathematical giant. So it is surprising that Pr. X is withholding such an important document, ostensibly not to "spoil" the image of his illustrious (biological) ancestors.
But I have a conjecture about the true reason why this ("illegimate") scion is not making this priceless document public. According to rumors, Gauss describes, in this newly found diary, the true method that lead him, when he was a mere 7-years-old lad, to figure out, in less than a minute, that
1+2+ ... +100=5050 .
In 2006, the great Amer. Sci. computing columnist, Brian Hayes, has already debunked the conventional belief that Gauss did it by the "method of bridges",
(1+100)+(2+99)+ ... + (49+52) +(50+51)= (101)(50)=5050 ,
or, almost equivalently, by adding it to its reverse
1+ 2+...+ 100 + 100+ 99+ ...1 = 101+101+...+101=(101)(100) ,
and then dividing by 2. Brian Hayes speculated about some alternative methods, none of them turned out to be the right one.
According to the diary, written by Gauss shortly before he died (and apparently entrusted to his "bastard" daughter that he must have kept in touch with), he described how he actually figured out the right answer.
He reasoned as follows:
1=1=(1)(2)/2 ,
1+2=3=(2)(3)/2 ,
1+2+3=6=(3)(2)=(3)(4)/2 .
Then, being the bright lad that he was, he detected a pattern, and made a conjecture
1+2+...+n=n(n+1)/2 ,
without proving it by "complete mathematical induction", (or by the "method of bridges" that was attributed to him by later unreliable historians like E.T. Bell), after all, he was only seven-years-old!, he "plugged-in" n=100, and got the right answer.
We all know that in addition to being a great prover, Gauss was also a great experimental mathematician, conjecturing the prime number theorem almost a hundred years before it was proved "rigorously". The fact that his amazing feat, when he was seven-years-old, was done "only" via experimental math, using "incomplete induction", (and not the method of bridges [or, worse, the tedious "proof by complete mathematical induction" that we indoctrinate our "introduction to Proofs" students with]), only, enhances my image of Gauss as a true visionary, and Pr. X should not be embarrassed. Not bad for a seven-years-old child.
I am sure that the diary contains lots of other gems, both personal, and mathematical, and Pr. X owes it to us to make it public! And don't be embarrassed, Pr. X. Even though Gauss's argument is empirical, it is fully rigorous! As Gauss himself should have realized by the time he was eleven, since both
1+2+ ...+n and n(n+1)/2
are polynomials of degree 2, once you check that they are equal for three different values, they must be identically equal, so Gauss had a fully rigorous proof (without knowing it, at the time).
And if you ask me, seven-years-old Gauss's "empirical" approach is far superior to the "clever", but ad-hoc, "method of bridges", erroneously attributed to him by later historians. For example, if you try, to do
12+22+32+...+ 1002 ,
the method of bridges fails miserably, but Gauss's "experimental" method works!
EmptySum=0=0(1)(1)/6 ,
12=1=(1)(2)(3)/6 ,
12+22=5=(2)(3)(5)/6 ,
12+22+32=14=(3)(4)(7)/6 .
Now detect a pattern
12+22+32+ ...+ n2=n(n+1)(2n+1)/6 ,
and, bingo ,
12+22+32+ ...+ 1002=(100)(101)(201)/6= 338350 ,
and once again, this is a fully rigorous proof! (why?)
So Pr. X, PLEASE reconsider. Of course, the original copy should remain with you, but all you need to do is go to a scanner, and then upload it to your website (if it exists). If you don't have a scanner, then ask someone to make a photocopy, and I'll be more than glad to do the scanning, and post it in my own website, of course with due credit to you. If you feel that some parts are too "private", then you may copy only the mathematical parts, but please, you owe it to us to have open access to the gems of the "Prince of Mathematics". Please! Bitte Schön!!, S'il vous plait!!!