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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. }

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