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While I was looking for a gift for a child’s birthday, a math book fell into my hands. I am always fascinated when authors write about abstract scientific topics for children, whether it’s on Albert Einstein’s theories, the life of Marie Curie, technology, or space travel. But this particular book was different. It’s all about prime numbers—specifically twin primes. Danish author Jan Egesborg has endeavored to introduce children to one of the most stubborn open problems in number theory, which even the brightest minds have repeatedly failed to solve over the past 100-plus years: the twin prime conjecture.

As is so often the case in mathematics, the conjecture falls into the category of those that are easy to understand but devilishly hard to prove. Twin primes are two prime numbers that have a distance of two on the number line; that is, they are directly consecutive if you ignore even numbers. Examples include 3 and 5, 5 and 7, and 17 and 19. You can find a lot of twin primes among small numbers, but the farther up the number line you go, the rarer they become.

That’s no surprise, given that prime numbers are increasingly rare among large numbers. Nevertheless, people have known since ancient times that infinite prime numbers exist, and the prime number twin conjecture states that there are an infinite number of prime number twins, as well. That would mean that no matter how large the values considered, there will always be prime numbers in direct succession among the odd numbers.

Admittedly, translating these concepts for kids is not easy (which is why I have so much respect for Egesborg and his children’s book). Prime numbers (2, 3, 5, 7, 11, 13,…) are like the fundamental particles of the natural numbers. They are only divisible by 1 and themselves. All other natural numbers can be broken down into their prime divisors, which makes prime numbers the basic building blocks of the mathematical world.

A Proof from Antiquity

Mathematics has an unlimited number of prime number building blocks. Euclid proved this more than 2,000 years ago with a simple thought experiment. Suppose there were only a finite number of prime numbers, the largest being p. In this case, all prime numbers up to p could be multiplied together.

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The best high-protein snacks can serve a vital role: Meeting your protein goals can be a challenge if you go hard at the gym. Without getting enough protein, you won’t be able to maintain or build muscle. Hopefully, you’re already getting protein at every meal, but it can still be tough to hit your number, and the right snacks are can helps The key is having some of the macronutrient readily accessible, so here are 10 high-protein snack ideas—all recommended by registered dietitians.

What’s Considered a “High Protein” Snack, Anyway?

You already know that protein is important, but maybe things get fuzzy when it comes to how much you should aim to get each day. Registered dietitian and From Burnout to Balance author Patricia Bannan, MS, RDN, says that while individual needs vary based on one’s weight, age, and activity level, a good general rule to follow is consuming .8 grams of protein per kilogram of bodyweight and between 1.2 to 2.0 grams per kilogram of bodyweight if you’re working out regularly or super active.

Take someone who weighs 180 pounds, or 82 kilograms. For the minimum protein requirement (0.8 grams per kg), multiply that by 0.8, which comes out to 65 grams.

While hitting your daily protein goal is important, registered dietitian Andy De Santis, RD, suggests not to get too obsessive. “A protein target is important, but so is having an overall balanced diet,” he says. If you’re eating the same high-protein foods continuously instead of a wider range of nutrient-rich foods just to stay under a certain calorie budget, De Santis says it’s time to take a step back and reevaluate. What’s most important, he says, is eating a wide variety of nutrient-rich foods so you’re getting a broad mix of everything the body needs.

With all of this in mind, what exactly is considered a high-protein snack? Registered dietitian Kristin Kirkpatrick, RD, says this depends on someone’s individual protein goals, but she considers a high-protein snack to be something with between 10 and 25 grams of protein. Any less than that, wouldn’t be considered “high protein” and any more than that is veering into meal territory, not a snack.

When Is a Good Time to Eat a High-Protein Snack?

“What’s most important is hitting your daily protein target. From there, you can consider how you’re meeting that target throughout the day,” De Santis says, emphasizing that this is where high-protein snacks come in handy. “High-protein snacks can help close the gap between the protein people are getting at their meals and their daily goal,” he says.

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Photographs: Getty Images; Collage: Armando Zaragoza

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If you’ve experienced this phenomenon, you’ve had a prospective memory lapse.

Memory usually means remembering things that have already happened. But prospective memory is the ability to remember to do something in the future – such as stopping to get milk on the way home from work, calling your mom on her birthday, or remembering to take your casserole out of the oven. Sometimes, errors lead to heartbreaking results – such as forgetting to take your toddler out of the car on a hot day.

I am a clinical neuropsychologist and a professor of psychology and neuroscience. For the past 30 years, my research has focused on this phenomenon, measuring prospective memory and looking for treatments to help those having problems.

Carrying out future intentions

Prospective memory is the ability to remember to remember, or to remember to carry out a future intention.

A future intention can be retrieved in two different ways. One is in response to something in your environment, such as a sight or a sound, that serves as a cue to do an intended action. Researchers like me call this an “event.” For instance, you see your co-worker and remember you have a message for them; you hear the timer go off and remember to turn off the sprinklers.

The other way is in response to time. Your dental appointment is scheduled for 2 p.m., but you plan to exercise for 30 minutes first.

One possible explanation for how people retrieve a prospective memory is known as the multiprocess theory.

According to this theory, sometimes you have to put forth effort to remember, such as checking the time repeatedly until it gets to be 2 p.m. Other times the memories come to you without effort, such as when you hear an alarm. As the above examples suggest, memories that have time-based cues, like 2 p.m., are usually more difficult to remember than those that have event-based cues, such as an alarm.

Brain regions at play

Older adults tend to lose prospective memory as they age. This may be because of brain changes in the aging prefrontal cortex.

But it’s not all bad news. Older adults actually seem to do better than younger people in some situations when they are asked to remember things in their daily lives. We call this the age-prospective memory paradox.

The part of the brain that seems most responsible for prospective memory is an area of the frontal lobes, referred to as Brodmann’s area 10. This area is involved with holding information in mind while doing a different simultaneous task.

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Ready to put together an arms-and-shoulders workout? I’ve already given you the lowdown on the best bicep exercises, the best tricep exercises, and best shoulder exercises. Today I’ll give you a formula to combine them, and it’s customizable so you can swap in your favorites if you don’t love mine.

The structure

We’re going to start with a heavy(ish) compound exercise that works your arms and shoulders together. Then we’ll move on to moderately heavy exercises, starting with the ones that didn’t get hit very hard on the compound. We’ll superset exercises where we can, to save time. It will look something like this: 

1. Shoulder press of your choice – 5 sets, heavy, anywhere between 3-8 reps, with 2 minutes of rest between sets.
2. (optional) Rear delt work – 3 sets of 8-12 reps
3. Bicep/tricep superset – 3 sets of 8-12 reps each 
4. Shoulder superset – 3 sets of 10+ reps each
5. (optional) Light bicep/tricep superset – 3 sets of 12-15 reps each

For a 30 minute workout, skip the optional sections and just do a shoulder press, bicep/tricep superset, and a shoulder superset. The first 15 minutes will be spent on the shoulder press and the rests between sets; the other components you can speed through with basically no rest. 

To get more work in, take some extra time and do the optional sections. If you try that and find you’re recovering well, it’s fine to add more bi/tri supersets or double up on any of the components of the workout that you’d like to spend more time on.

How this fits into your week: Once a week is okay, but this workout should ideally be done twice per week. It’s also fine to do this workout once, and a different upper-body workout another time. 

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Credit: Khakimullin Aleksandr/Shutterstock

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Wonderful idea… in the election of 2024 perhaps!

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In politics, opprobrious language and conduct, plus outright lying should be condemned, against the law, and severely punished!

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