# Linear support for the prime number sequence and the first and second Hardy-Littlewood conjectures

## DOI:

https://doi.org/10.7146/math.scand.a-143425## Abstract

Servais and Grün used results about linear support for the prime number sequence to obtain upper bounds on the smallest prime in odd perfect numbers. This was extended by Cohen and Hendy who proved that for every $n \in \mathbb {N}$ there exists an integer $b_n$ such that $p_i \ge p_1 + ni - b_n$ for any sequence $(p_i)$ of odd primes, and found minimal values of $b_n$ for $3 \le n \le 5$, and conjectured minimal values for $6 \le n \le 10$. We give a new proof of the existence of $b_n$ and the values for $3 \le n \le 5$. We also show that if we assume that the second Hardy-Littlewood conjecture, $\pi (a+b) \le \pi (a) + \pi (b)$ for $a, b \ge 2$, is true, then we can in a finite number of steps determine numbers, $T_n$, that give quite close bounds for the values of $b_n$, namely $T_n \le b_n \le T_n + n - 2$, and determine the values of $T_n$ for $6 \le n \le 20$.

We also consider the question of whether the values of $b_n$ can be replaced by smaller numbers if we assume that $p_1 > 3$. We will show that if we assume that the first Hardy-Littlewood conjecture is true, then we can determine the minimum such values, $a_n$, for $3 \le n \le 5$. We also determine some lower bounds for $a_n$ for $n \ge 6$. It is well-known that the two Hardy-Littlewood conjectures are mutually exclusive, but we never use both conjectures simultaneously.

These results give us the upper bound $p_1<\frac {n}{2^n-1}t+b_n$ on the smallest prime in odd perfect numbers, where $t$ is the number of distinct primes dividing the odd perfect number. This bound was also found by Cohen and Hendy, but we improve the constants $b_n$.

## References

Anonymous, Smallest prime k-tuplets, http://goodprimes.eu5.org/Txt/ktmin.txt.

Aslaksen, H., and Kirfel, C., Linear support for the prime number sequence with slope 4 or 5 (in preparation).

Betcher, J. T., and Jaroma, J. H., An extension of the results of Sevais and Cramer on odd perfect and odd multiply perfect numbers, Amer. Math. Monthly 110 (2003), no. 1, 49–52. https://doi.org/10.2307/3072344

Cohen, G. L., On increasing sequences of odd primes, Austral. Math. Soc. Gaz. 3 (1976), 84–85.

Cohen G. L., and Hendy, M. D., Polygonal supports for sequences of primes, Math. Chronicle 9 (1980), 120–136. (Addendum: On odd multiperfect numbers, Math. Chronicle 10 (1981), 57–61.)

Grün, O., Über ungerade vollkommene Zahlen, Math. Z. 55 (1952), no. 3, 353–354. https://doi.org/10.1007/BF01181133.

Hardy, G. H., and Littlewood, J. E., Some problems of `Partitio Numerorum'. III. On the expression of a number as a sum of primes, Acta Math. 44 (1923), no. 1, 1–70. https://doi.org/10.1007/BF02403921

Hardy, G. H., and Wright, E. M., An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979

Montgomery, H. L., and Vaughan, R. C., The large sieve, Mathematika 20 (1973), no. 2, 119–134. https://doi.org/10.1112/S0025579300004708.

Nielsen, P. P., Odd perfect numbers, Diophantine equations, and upper bounds, Math. Comp. 84 (2015), no. 295, 2549–2567. https://doi.org/10.1090/S0025-5718-2015-02941-X

Ochem, P., and Rao, M., Odd perfect numbers are greater than $10^1500$, Math. Comp. 81 (2012), no. 279, 1869–1877. https://doi.org/10.1090/S0025-5718-2012-02563-4

Richards, I., On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem, Bull. Amer. Math. Soc. 80 (1974), 419–438. https://doi.org/10.1090/S0002-9904-1974-13434-8

Rosser, J. B., and Schoenfeld, L., Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. http://projecteuclid.org/euclid.ijm/1255631807

Segal, S. L., On $pi (x + y) le pi (x) + pi (y)$, Trans. Amer. Math. Soc. 104 (1962), no. 3, 523–527. https://doi.org/10.2307/1993801

Servais, C., Sur les nombres parfaits, Mathesis, 8 (1888), 92–93.

Sutherland, A. V., Narrow admissible tuples, https://math.mit.edu/ primegaps/.

Zelinsky, J., On the total number of prime factors of an odd perfect number, Integers 21 (2021), Paper No. A76, 55 pp.

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*MATHEMATICA SCANDINAVICA*,

*130*(2). https://doi.org/10.7146/math.scand.a-143425