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Leonhard Euler
was born at Bāle on April 15, 1707, and died at St. Petersburg on
September 7, 1783. he was the son of a Lutheran minister who had
settled at Bāle, and was educated in his native town under the
direction of John Bernoulli, with whose sons Daniel and Nicholas
he formed a lifelong friendship. When, in 1725, the younger
Bernoullis went to Russia, on the invitation of the empress, they
procured a place there for Euler, which in 1733 he exchanged for
the chair of mathematics, then vacated by Daniel Bernoulli. The
severity of the climate affected his eyesight, and in 1735 he lost
the use of one eye completely. In 1741 he moved to Berlin at the
request, or rather command, of Frederick the Great; here he stayed
till 1766, when he returned to Russia, and was succeeded at Berlin
by Lagrange. Within two or three years of his going back to St.
Petersburg he became blind; but in spite of this, and although his
house, together with many of his papers, were burnt in 1771, he
recast and improved most of his earlier works. He died of apoplexy
in 1783. He was married twice.
In
the first place, he wrote in 1748 his Introductio in
Analysin Infinitorum, which was intended to serve as an
introduction to pure analytical mathematics. This is divided into
two parts.
If p is an odd prime number, then the formula 
can always be divided by p.
PROOF
In place of 2 write 1+1 and get
The number of terms in this series is p, and is thus
odd. Moreover, any term which has the form of a fraction gives an
integer number since the numerator is always divisible by its
denominator. Cutting off the first term, 1, of this series gives
The number of terms of this series is p-1, and is thus
even. Thus, collecting pairs of terms into one sum, so the number
of terms will be half as many, gives
Since p is an odd number, the last term of this series
will be
It now becomes visible that each term is divisible by p,
since p is a prime number larger than any factor in the
denominator, so division by the denominator cannot eliminate the
factor. For that reason, if p is an odd prime number, then
may always be divided by it. Q. E. D.
In another way, if
can be divided by a prime number p, so also, in turn, will
be its double,  ,
and so (see note)
Cutting off the first and last terms of this series gives
Looking at this series, each term is divisible by p,
since p is a prime number. Thus always
can be divided by p, unless p=2. Q. E. D.
Since
can be divided by an odd prime, it is easily known that p
divides any formula 
, where m denotes any integer whatsoever. Thus the prime number p
can divide the following forms 
, 
, 
etc. Therefore the truth of this theorem is proved in general for
all cases where a is any power of 2 and p is a prime
number except two.
We will now prove the following
THEOREM
Denoting by p any prime number except 3, the formula 
can always be divided by p.
PROOF
If
is divisible by a prime number other than 3, then 
is divisible by it, and conversely. It is true then that
in which series each term except the first and the last is
divisible by p, if p is a prime number. Therefore
the formula 
is divisible by p, which is equal to this:
And
is always divisible by the prime number p, and so also is .
And so
is always divisible by p, as long as p is a prime
number other than 3. Q. E. D.
In this manner it is possible to progress from a given value a
to the value one greater. But by this proof one may produce a
general theorem by putting together carefully larger and larger
numbers, which leads to the following
THEOREM
Denoting by p a prime number, if 
can be divided by p, then so also can the formula  .
PROOF
is resolved into a series in the usual way to get
in which series each term is divisible by p except the
first and the last, if p is a prime number. For that
reason, 
can be divided by p, and this formula is the same as 
. And
can be divided by p, by hypothesis, and therefore 
can be also. Q. E. D.
Therefore, with the knowledge that ap-a can be
divided by the prime number p, it admits the division of
by p, and it follows 
, and then 
and generally 
to be divisible by p. Putting a=2, and we have
already shown that
can be divided by p, and it follows that the formula 
ought to admit division by p, whatever number is
substituted in place of b.
Therefore p divides
, and consequently 
, unless a=p or a is a multiple of p. |
