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The proliferation of probable prime tests in recent years has produced a plethora of definitions with the word " pseudoprime " in them. Examples include pseudoprimes, Euler pseudoprimes, strong pseudoprimes, Lucas pseudoprimes, strong Lucas pseudoprimes, extra strong Lucas pseudoprimes and Perrin pseudoprimes. Though these tests represent a wealth of ideas,… (More)

- Jon Grantham
- 2008

Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most 1 4 of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius automor-phism. The test, along with a xed number of trial divisions, ensures that a composite n will pass for less than 1 7710 of the… (More)

- JON GRANTHAM
- 1995

We define g(n) to be the maximal order of an element of the symmetric group on n elements. Results about the prime factorization of g(n) allow a reduction of the upper bound on the largest prime divisor of g(n) to 1.328 V" log"-Let S " be the symmetric group on « letters. Definition. g(n) = max {ord(tr) | a e Sn}. The first work on g(n) was done by Landau… (More)

Adrenomedullin (ADM), a potent natriuretic and vasorelaxing peptide with inotropic properties, is elevated in plasma in human and experimental congestive heart failure (CHF). Recent studies suggest that angiotensin II stimulates ADM production and secretion from cardiac myocytes and fibroblasts. In the present study, we investigated cardiac ADM in… (More)

Interventricular septal hematoma is a rare complication of retrograde chronic total occlusion (CTO) percutaneous coronary interventions (PCI) with a typically benign course. Here we report two cases of interventricular septal hematoma and coronary-cameral fistula development after right coronary artery (RCA) CTO-PCI using a retrograde approach. Both were… (More)

BACKGROUND
Intravascular imaging can facilitate chronic total occlusion (CTO) percutaneous coronary intervention.
METHODS AND RESULTS
We examined the frequency of use and outcomes of intravascular imaging among 619 CTO percutaneous coronary interventions performed between 2012 and 2015 at 7 US centers. Mean age was 65.4±10 years and 85% of the patients… (More)

BACKGROUND
High success rates are achievable for chronic total occlusion (CTO) percutaneous coronary intervention (PCI) using the hybrid approach, but periprocedural complications remain of concern. Although scores estimating success and efficiency in CTO PCI have been developed, there is currently no available score for estimation of the risk for… (More)

We investigate the problem of finding integers k such that appending any number of copies of the base-ten digit d to k yields a composite number. In particular, we prove that there exist infinitely many integers co-prime to all digits such that repeatedly appending any digit yields a composite number.

- Jon Grantham
- 2006

This paper proves the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. The result uses ingredients of the proof of the infinitude of Carmichael numbers, along with zero-density estimates for Hecke L-functions.