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A NOTE ON RAMANUJAN'S CONJECTURES REGARDING `MERSENNE'S PRIMES'
P. G. Brown
In the so-called Lost Notebook of Ramanujan [1], pp. 259-260,
Ramanujan says that when 2^p-1 is a prime, p may be termed a
Mersenne's Prime. He then makes the following remarkable statements.
(1) \lq\lq All Mersenne's primes are either of the form a^2+ab+b^2 or of
the form a^2+b^2. Then since a number of the form 12n-1 cannot be
expressed in any one of the above two forms, we infer that
(2) A Mersenne's prime is never of the form 12n-1. Thus for example
2^{11}-1, 2^{23}-1, 2^{47}-1, 2^{59}-1, 2^{71}-1, 2^{83}-1, 2^{107}-1, 2^{131}-1,
2^{167}-1, 2^{179}-1, 2^{191}-1, 2^{227}-1, 2^{239}-1, 2^{251}-1, &c
should be composite numbers. Hence we may divide all Mersenne's primes into
two classes, one comprising primes that can be expressed as a^2+ab+b^2
and the other containing primes that cannot be expressed as a^2+ab+b^2.
(3) Hence the Mersenne's primes of the 1st-class except 1 and 3 are
of the form 6n+1, while those of the 2nd except 2 are of the
form 12n+5. Thus we have,
\align
&\text {No.s of the 1st class:- } 1,3,7,13,19,31,61,127,\text {&c}\\
&\text {No.s of the 2nd class:- } 2,5,17,89,257,\text {&c}
align
(4) Theorem.
If P be any prime, and p any odd prime and if either of
\frac^p-1}-1} or
\frac{ P^p-1}(P-1)} happens to be a prime,
then that prime will be a Mersenne's prime of the 1st class.
As a particular case we have when p = 3,
(5) If P be any prime and if either of P^2+P+1 or
\frac^2+P+1}{3} happens to be a prime,
then that prime will be a Mersenne's prime of the 1st class.
(6) If p be a Mersenne's prime then 2^p-1 will be a Mersenne's prime
of the 1st class. As examples of (5) and (6) we have
{1^2+1+1=3};
{2^2+2+1=7};
{3^2+3+1=13};
{5^2+5+1=31};
{\frac{7^2+7+1}{3}=19};
{(11^2+11+1=133 \text {composite})};
{\frac{13^2+13+1}{3}=61};
{17^2+17+1=307};
{\frac{19^2+19+1}{3}=127};
and so on. Again
{2^2-1=3}; hence {2^3-1=7} a prime;
hence
{2^7-1=127} a prime;
hence {2^{127}-1}
is a prime.
{2^5-1=31}; hence {2^{31}-1} is a prime.
(7) From (3) we can infer that the number of Mersenne's primes of the 2nd
class is always about \frac{1}{2} of the
number of those of the 1st class. There may be a general theorem like (4)
for the Mersenne's primes of the 2nd class of which the particular case
analogous to (6) will be,
(8) If 2^p+1 be a prime, then 2^p+1 will be a Mersenne's prime of the
2nd class. Thus for example we have
{2+1=3}; hence 2^3-1 is a prime; {2^2+1=5} hence {2^5-1} is a prime;
{2^4+1=17} hence {2^{17}-1} is a prime; {2^8+1=257}; hence {2^{257}-1} is a prime and so on.
(9) Mersenne's primes of the 2nd class are always of the form (2^a)^2+(4b+1)^2 where a assumes all
integral values, 0, 1, 2, 3 &c without an exception, b is a positive integer
including 0, 4b+1 is never greater than 2^a and for every value of a,
there is at least one value of b. Thus we have, when a = 0, b = 0
and hence 2^2 - 1 is a prime; when a = 1, b must be 0 hence 2^5 - 1
is prime; when a = 2, b must be 0 hence 2^{17} - 1 is a prime; when
a = 3, b may be 0 or 1, but when b = 0, (2^8)^2 + 1 becomes
composite, hence b must be 1 since (2^3)^2+5^2 is a prime, hence
2^{89} - 1 is a prime.
(10) Another theorem analogous to (8) is, if {2^p+1} is a prime
then {2^{2^p}+1} is also a prime."
Note Ramanujan's initial definition of a Mersenne's prime to be
the (necessarily) prime index and not the prime 2^p-1 itself.
Unfortunately all the conjectures above are false.
(1) False. 107 is not of the form a^2+b^2 nor of the form
a^2+b^2 + ab,
however it is a Mersenne's prime (sic), since 2^{107}-1 is prime.
(2) False. 107 will again do as a counter-example.
(3) Not important given the falsehood of (1) and (2).
Also note that 2^{257}-1
is NOT prime as claimed in the list.
(4) False: e.g. with P=7, p=5 the first of the given formulae gives
2801 which is prime, but 2^{2801}-1 is composite. P=31, p=3 in the second
formula gives the prime 331, but 2^{331}-1 is composite.
(5) False: e.g. P=17 in the first formula gives 307, a prime,
but 2^{307} -1
is composite, so 307 is not a Mersenne's prime. Also, P=31 in the second
formula gives 331 again which is not a Mersenne's prime.
(6) False: e.g. 13 is a Mersenne's prime but 2^{13}-1 = 8191 is not.
(7) Not important given (1), (2) and (3).
(8) False. As mentioned above 2^{257}-1 is NOT prime.
(9) {False: e.g. $4253\equiv 5$ mod 12 and $2^{4253}-1$ is prime, but the
(unique up to order) sum of two squares representation of 4253 is
$53^2 + 38^2$ and neither summand is a power of 2.
Also if $a=4$, then $b=0,1,2$ or 3
and so $(2^a)^2 + (4b+1)^2 = 257,281,337,425$ respectively and none of these
numbers are Mersenne's primes.}
(10) {False: e.g. $2^8+1$ is prime, but $2^{2^8} +1 $ is composite.}
Clearly even the great Ramanujan had his `bad days'.
References
[1] {S. Ramanujan The Lost Notebook and Other Unpublished Papers , 1988,
Narosa, New Delhi. }
School of Mathematics
University of New South Wales
Sydney 2052
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