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The number of verified 64-bit numbers per second for various size of base bit of table S in the GPU implementation for the convergence of the Collatz conjecture
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The main contribution of this paper is to present an implementation that performs the exhaustive search to verify the Collatz conjecture using a GPU. Consider the following operation on an arbitrary positive number: if the number is even, divide it by two, and if the number is odd, triple it and add one. The Collatz conjecture asserts that, startin...
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... we find the optimal size of bits for table S. Figures 5 and 6 show the number of verified numbers per second for various base bit of table S in the GPU and CPU implementations, respec- tively. According to the both graphs, when the base bit is larger, the number is larger because the number of non-mandatory numbers is larger for larger base bit as shown in Table 3. Due to the size limitation, more than 37 bits for table S cannot be stored in the global memory in the GPU and the main memory in the CPU, respectively. ...
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La conjecture de Syracuse, introduite par Lothar Collatz et ouverte depuis 1928 est encore appelée conjecture de Collatz, conjecture d'Ulam, conjecture tchèque ou problème 3x+1.
Le but de cet article est de démontrer cette belle conjecture dont Paul Erdos a dit " les mathématiques ne sont pas encore prêtes pour de tels problèmes " .
Citations
... Also, in 2017, Honda et al. [10] claimed they can check 2 40.25 numbers per second on the GPU. Their program is, however, only able to verify 64-bit numbers. ...
This article presents our project, which aims to verify the convergence of the Collatz problem computationally. As a main point of the article, we introduce a new result that pushes the limit for which the problem is verified up to 1.5 × 270. We present our baseline algorithm and then several sub-algorithms that bring some acceleration. The total acceleration from the first algorithm we used on the CPU to our best algorithm on the GPU is 1 335×. We further distribute individual tasks to thousands of parallel workers running on several European supercomputers. Besides the convergence verification, our program also checks for path records during the convergence test. We found four new path records.
... In 2017, the yoyo@home project [1] checked for convergence all numbers up to 10 20 ≈ 2 66.4 . The work of Honda et al. [7] claims that they can check 2 40.25 numbers per second for the convergence (the first group), or 2 29.9 numbers per second for the delay, both on GPU. Their programs are however only able to verify 64-bit numbers (which have been known to converge due to the yoyo@home project). ...
... We experimented with many sieve sizes and came to the conclusion that the sieve size 2 34 is optimal for our CPU implementation, whereas the sieve size 2 24 is optimal for GPUs. We are aware that other authors have used even larger sieves (and therefore have reached higher performance), e.g., the sieve of the size 2 37 in [7]. However, such a sieve is absolutely impractical since it occupies 16 gigabytes of memory. ...
... Additionally, the program sums all the s for all n inside a particular work unit as proof of work so that the results can be independently verified. These data were not collected in [7], whereas our program is focused on practical use. A comparison of our program with competing programs is given in Table 1. ...
This article presents a new algorithmic approach for computational convergence verification of the Collatz problem. The main contribution of the paper is the replacement of huge precomputed tables containing O(2^N) entries with small lookup tables comprising just O(N) elements. Our single-threaded CPU implementation can verify 4.2 × 10^9 128-bit numbers per second on Intel Xeon Gold 5218 CPU computer, and our parallel OpenCL implementation reaches the speed of 2.2 × 10^11 128-bit numbers per second on NVIDIA GeForce RTX 2080. Besides the convergence verification, our program also checks for path records during the convergence test.
... In other words, the conjectures say that the Collatz sequence does not contain nontrivial cycles and also does not grow indefinitely. In the past and present many authors [1][2][3][4][5][6][7][8][9][10][11] try to establish the validity of the conjecture in affirmatively but till today there is no theoretical proof for the same and there is no counter example to disprove the conjecture as well. ...
Lothar Collatz had proposed in 1937 a conjecture in number theory called Collatz conjecture. Till today there is no evidence of proving or disproving the conjecture. In this paper, we propose an algorithmic approach for verification of the Collatz conjecture based on bit representation of integers. The scheme neither encounters any cycles in the so called Collatz sequence and nor the sequence grows indefinitely. Experimental results show that the Collatz sequence starting at the given integer , oscillates for finite number of times, never exceeds 1.7 times (scaling factor) size of the starting integer and finally reaches the value 1. The experimental results show strong evidence that conjecture is correct and paves a way for theoretical proof.
... Current computational capabilities have allowed confirming the conjecture for very large numbers. For example, Honda et al. (2017) introduced a GPU-based method to verify the Collatz algorithm. The authors could verify 1.31e12 64-bit numbers per second. ...
Cryptocurrencies such as Bitcoin rely on a proof-of-work system to validate transactions and prevent attacks or double-spending. A new proof-of-work is introduced which seems to be the first number theoretic proof-of-work unrelated to primes: it is based on a new metric associated to the Collatz algorithm whose natural generalization is algorithmically undecidable: the inflation propensity is defined as the cardinality of new maxima in a developing Collatz orbit. It is numerically verified that the distribution of inflation propensity slowly converges to a geometric distribution of parameter 0.714 ≈(π−1)/3 as the sample size increases. This pseudo-randomness opens the door to a new class of proofs-of-work based on congruential graphs.
Cryptocurrencies like Bitcoin rely on a proof-of-work system to validate transactions and prevent attacks or double-spending. A new proof-of-work is introduced which seems to be the first number theoretic proof-of-work unrelated to primes: it is based on a new metric associated to the Collatz algorithm whose natural generalization is algorithmically undecidable: the inflation propensity is defined as the cardinality of new maxima in a developing Collatz orbit. It is numerically verified that the distribution of inflation propensity slowly converges to a geometric distribution of parameter $0.714 \approx \frac{(\pi - 1)}{3}$ as the sample size increases. This pseudo-randomness opens the door to a new class of proofs-of-work based on congruential graphs.
Cryptocurrencies like Bitcoin rely on a proof-of-work system to validate transactions and prevent attacks or double-spending. Reliance on a few standard proofs-of-work such as hashcash, Ethash or Scrypt increases systemic risk of the whole crypto-economy. Diversification of proofs-of-work is a strategy to counter potential threats to the stability of electronic payment systems. To this end, another proof-of-work is introduced: it is based on a new metric associated to the algorithmically undecidable Collatz algorithm: the inflation propensity is defined as the cardinality of new maxima in a developing Collatz orbit. It is numerically verified that the distribution of inflation propensity slowly converges to a geometric distribution of parameter $0.714 \approx \frac{(\pi - 1)}{3}$ as the sample size increases. This pseudo-randomness opens the door to a new class of proofs-of-work based on congruential graphs.
Cryptocurrencies like Bitcoin rely on a proof-of-work system to validate transactions and prevent attacks or double-spending. Reliance on a few standard proofs-of-work such as hashcash, Ethash or Scrypt increases systemic risk of the whole crypto-economy. Diversification of proofs-of-work is a strategy to counter potential threats to the stability of electronic payment systems. To this end, another proof-of-work is introduced: it is based on a new metric associated to the algorithmically undecidable Collatz algorithm: the inflation propensity is defined as the cardinality of new maxima in a developing Collatz orbit. It is numerically verified that the distribution of inflation propensity slowly converges to a geometric distribution of parameter 0.714 ≈ (π−1) / 3 as the sample size increases. This pseudo-randomness opens the door to a new class of proofs-of-work based on congruential graphs.
