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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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    University of New South Wales
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